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Advanced Study Material for Class 10 Mathematics Chapter 14 Probability
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Class 10 Mathematics Chapter 14 Probability Notes and Questions
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1. On one page of a telephone directory, there were 200 telephone numbers. The frequency distribution of their unit place digit (for example, in the number 25828573, the unit place digit is 3) is given in below table:
Without looking at the page, the pencil is placed on one of these numbers, i.e., the number is chosen at random. What is the probability that the digit in its unit place is (i) an odd number (ii) a prime number and (iii) a number greater than 4.?
2. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). neither divisible by 5 nor 10.
n). neither divisible by 2 nor 5.
o). neither divisible by 3 nor 5.
p). a perfect cube number.
q). a prime number
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
3. Cards are marked with numbers 4, 5, 6, …….50 are placed in the box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). neither divisible by 5 nor 10.
n). neither divisible by 2 nor 5.
o). neither divisible by 3 nor 5.
p). a perfect cube number.
q). a prime number
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
4. Cards are marked with numbers 13, 14, 15, …….60 are placed in the box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
5. There are 30 cards numbered from1 to 30. One card is drawn at random. Find the probability of getting the card with
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
6. A box contains 25 cards numbered from 1 to 25. A card is drawn from the box at random. Find the probability of getting the card with
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
7. A box contains 19 balls bearing numbers 1,2,3, …. 19 respectively. A ball is drawn at random from the box, Find the probability that the number on the ball is
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
8. A box contains 20 balls bearing numbers 1,2,3, …. 20 respectively. A ball is drawn at random from the box, Find the probability that the number on the ball is
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
9. 15 cards numbered 1, 2, 3, 4,……. 14, 15 are put in a box and mixed thoroughly. A man draws a card at random from the box. Find the probability that the number on the card is
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
10. Tickets numbered 2, 3, 4, 5, …..100, 101 are placed in a box and mixed thoroughly. One ticket is drawn at random from the box. Find the probability that the number on the ticket is
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
11. Cards are marked with numbers 5, 6, 7, …….50 are placed in the box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting.
a). a two-digit number
b). a perfect square number
c). a number divisible by 5.
d). a number divisible by 2 or 3.
e). a number divisible by 2 and 3.
f). a number divisible by 7.
g). a number multiple of 8.
h). a two digit number divisible by 5.
i). a two digit number divisible by 2.
j). a two digit number divisible by 3.
k). a two digit number divisible by 4.
l). a two digit number perfect square.
m). a perfect cube number.
n). a prime number.
o). neither divisible by 5 nor 10.
p). neither divisible by 2 nor 5.
q). neither divisible by 3 nor 5.
r). a two digit prime number.
s). an even prime number.
t). a number is not divisible by 5.
u). a number is not divisible by 3.
v). a number is not divisible by 2 and 3.
SHORT ANSWER TYPE QUESTIONS
Question. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball.
Answer: Total number of balls \( = 4 + 6 = 10 \)
\( \Rightarrow \) All possible outcomes \( = 10 \)
Since, number of black balls \( = 6 \)
\( \therefore \) Number of favourable outcomes \( = 6 \)
\( \Rightarrow P(E) = \frac{6}{10} \text{ or } \frac{3}{5} \).
Question. A die is thrown once. Find the probability of getting a number less than 3.
Answer: Numbers on the faces are 1, 2, 3, 4, 5 and 6.
\( \therefore \) Number of possible outcomes \( = 6 \)
Numbers less than 3 are 1 and 2.
\( \Rightarrow \) Number of favourable outcomes \( = 2 \)
\( \therefore P(E) = \frac{2}{6} \text{ or } \frac{1}{3} \).
Question. A die is thrown once. Find the probability of getting a number greater than 5.
Answer: Total number of possible outcomes \( = 6 \)
Since only one number i.e., 6 is greater than 5
\( \therefore \) Favourable number of outcomes \( = 1 \)
\( \Rightarrow P(E) = \frac{1}{6} \).
Question. Find the probability of obtaining 7 on a single toss of one die.
Answer: Numbers marked on a die are:
1, 2, 3, 4, 5, 6
\( \therefore \) There are six different possible outcomes.
But none of these outcomes would produce a 7.
\( \Rightarrow \) Favourable outcome \( = 0 \)
\( \therefore P_{(7)} = \frac{0}{6} = 0 \)
When an event cannot possibly succeed, we say it is an impossible event and probability of an impossible event is zero.
i.e. \( P_{\text{(impossible event)}} = 0 \)
Question. Cards bearing numbers 3 to 20 are placed in a bag and mixed thoroughly. A card is taken out from the bag at random. What is the probability that the number on the card taken out is an even number?
Answer: Total number of cards (3 to 20) \( = 18 \)
\( \therefore \) Number of possible outcomes \( = 18 \)
Since cards having even numbers (4, 6, 8, 10, 12, 14, 16, 18 and 20) are 9,
\( \therefore \) Number of favourable outcomes \( = 9 \)
\( \therefore P(E) = \frac{9}{18} \text{ or } \frac{1}{2} \).
Question. Two friends were born in the year 2000. What is the probability that they have the same birthday?
Answer: Since the year 2000 was a leap year,
\( \therefore \) Total number of days in the year \( = 366 \)
\( \because \) They have the same birthday.
\( \therefore \) Number of favourable outcomes \( = 1 \)
\( \Rightarrow P(E) = \frac{1}{366} \).
Question. A box contains cards marked with numbers 5 to 20. A card is drawn from the bag at random. Find the probability of getting a number which is a perfect square.
Answer: \( \because \) Total number of cards \( = 16 \)
\( \therefore \) Possible outcomes are 16.
Since the numbers 9 and 16 are perfect numbers,
\( \Rightarrow \) Number of favourable outcomes \( = 2 \)
\( \therefore P(E) = \frac{2}{16} \text{ or } \frac{1}{8} \).
Question. Two dice are thrown at the same time. Find the probability of getting different numbers on the dice.
Answer: Since the two dice are thrown simultaneously.
\( \therefore \) Total number of outcomes \( = 6 \times 6 = 36 \)
Number of outcomes for getting same numbers on both dice \( = 6 \)
\( \Rightarrow P \text{(same numbers)} = \frac{6}{36} = \frac{1}{6} \)
Now, \( P \text{(different numbers)} + P \text{(same numbers)} = 1 \)
\( \Rightarrow P \text{(different numbers)} = 1 - P \text{(same numbers)} \)
\( = 1 - \frac{1}{6} = \frac{5}{6} \).
Question. Two dice are thrown at the same time. Find the probability of getting same number on both dice.
Answer: Total number of outcomes \( = 6 \times 6 = 36 \)
\( \therefore \) Following are the outcomes that have same number on both dice are:
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6)
\( \therefore \) Favourable outcomes \( = 6 \)
\( \Rightarrow \) Required probability \( = \frac{6}{36} = \frac{1}{6} \).
Question. A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being not a blue ball.
Answer: Total number of balls \( = 10 + 5 + 7 = 22 \)
\( \therefore \) Number of possible outcomes \( = 22 \)
Since there are 5 blue balls.
\( \because \) Number of balls which are not blue \( = 22 - 5 = 17 \)
\( \therefore \) Favourable outcomes \( = 17 \)
\( \Rightarrow \) Required probability \( = \frac{17}{22} \).
Question. Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is less than 9.
Answer: Total number of possible outcomes \( = 6 \times 6 = 36 \)
\( \because \) The outcomes such that the product of numbers appearing on the faces is less than 9 are:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (4, 1), (4, 2), (5, 1) and (6, 1).
\( \therefore \) Number of favourable outcomes \( = 16 \)
\( \Rightarrow \) Required probability \( = \frac{16}{36} = \frac{4}{9} \).
Question. An integer is chosen between 0 and 100. What is the probability that it is divisible by 7?
Answer: \( \because \) Numbers between 0 and 100 are 99.
\( \therefore \) Total possible outcomes \( = 99 \)
Since following numbers are divisible by 7:
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91 and 98.
\( \therefore \) Favourable outcomes \( = 14 \)
\( \Rightarrow \) Required probability \( = \frac{14}{99} \).
Question. A letter of English alphabet is chosen at random. Determine the probability that the letter is consonant.
Answer: \( \because \) There are 26 letters of English alphabet
\( \therefore \) Number of possible outcomes \( = 26 \)
Since, there are 21 consonants of the English alphabets.
\( \therefore \) Favourable outcomes \( = 21 \)
\( \Rightarrow \) Required probability \( = \frac{21}{26} \).
Question. Cards with numbers 2 to 101 are placed in a box. A card is selected at random. Find the probability that the card has a square number.
Answer: Number of numbers between 2 to 101 are 100
\( \therefore \) Total number of possible outcomes \( = 100 \)
Since, the perfect numbers between 2 and 101 are:
4, 9, 16, 25, 36, 49, 64, 81 and 100
\( \therefore \) Number of favourable outcomes \( = 9 \)
\( \Rightarrow \) Required probability \( = \frac{9}{100} \).
Question. From a group of 2 boys and 3 girls, two children are selected at random. Find the probability such that at least one boy is selected.
Answer: Let \( B_1 \) and \( B_2 \) be two boys and \( G_1, G_2 \) and \( G_3 \) be the three girls
Since two children are selected at random,
\( \therefore \) Following are the possible groups:
\( B_1B_2, B_1G_1, B_1G_2, B_1G_3, B_2G_1, B_2G_2, B_2G_3, G_1G_2, G_1G_3, G_2G_3 \)
\( \therefore \) Total number of possible outcomes \( = 10 \)
Since, one boy is to be selected,
\( \therefore \) Favourable outcomes are:
\( B_1B_2, B_1G_1, B_1G_2, B_1G_3, B_2G_1, B_2G_2 \text{ and } B_2G_3 \).
\( \Rightarrow \) Number of favourable outcomes \( = 7 \)
\( \therefore \) Required probability \( = \frac{7}{10} \).
Question. A bag contains 7 red, 5 white and 3 black balls. A ball is drawn at random from the bag. Find the probability that the drawn ball is neither white nor black.
Answer: Total number of balls
\( = 7 + 5 + 3 \)
\( = 15 \)
\( \because \) Number of white balls \( = 5 \)
Number of black balls \( = 3 \)
\( \therefore \) Number of balls that are neither white nor black \( = 15 - [5 + 3] \)
\( = 15 - 8 = 7 \)
\( \therefore \) Required probability \( = \frac{7}{15} \).
Question. A box contains 20 cards, numbered from 1 to 20. A card is drawn from the box at random. Find the probability that the number on the drawn card is: (i) even (ii) multiple of 3.
Answer: Total numbers from 1 to 20 are 20
\( \therefore \) Number of possible events \( = 20 \)
(i) Even numbers are:
2, 4, 6, 8, 10, 12, 14, 16, 18 and 20
\( \therefore \) Number of favourable outcomes \( = 10 \)
\( \Rightarrow \) Probability of getting an even number \( = \frac{10}{20} = \frac{1}{2} \)
(ii) Since, multiples of 3 are:
3, 6, 9, 12, 15 and 18
\( \therefore \) Number of favourable outcomes \( = 6 \)
\( \Rightarrow \) Probability of getting a multiple of 3
\( = \frac{6}{20} = \frac{3}{10} \).
Question. Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is more than 9.
Answer: Following are the possible outcomes for two dice thrown simultaneously:
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
\( \therefore \) Total number of possible outcomes = 36
Following outcomes have a sum of more than 9:
(4, 6), (6, 4), (5, 5), (5, 6), (6, 5) and (6, 6)
i.e. Favourable outcomes = 6
\( \therefore \) The required probability = \( \frac{6}{36} \) or \( \frac{1}{6} \).
Question. In a bag-X, there are four cards numbered 1, 3, 5 and 7 respectively. In an another bag-Y, there are three cards numbered 2, 4 and 6 respectively. A card is drawn at random from each bag.
(a) Write all the possible outcomes.
(b) Find the probability that the sum of these two cards drawn is:
(i) 7 (ii) even (iii) more than 7
Answer: (a) There are 12 possible outcomes
| Bag A \( \rightarrow \) | 1 | 3 | 5 | 7 | 12 Possible outcomes | |
| Bag B \( \downarrow \) | ||||||
| 2 | 2, 1 | 2, 3 | 2, 5 | 2, 7 | ||
| 4 | 4, 1 | 4, 3 | 4, 5 | 4, 7 | ||
| 6 | 6, 1 | 6, 3 | 6, 5 | 6, 7 |
\( \Rightarrow \) Possible outcomes are (2, 1), (4, 1), (6, 1), (2, 3), (4, 3), (6, 3), (2, 5), (4, 5), (6, 5),
(2, 7), (4, 7) and (6, 7).
(b) (i) \( \because \) Only (6 + 1), (4 + 3) and (2 + 5) gives sum as 7
\( \therefore \) Possible outcomes = 3
\( \Rightarrow P_{\text{(Sum = 7)}} = \frac{3}{12} = \frac{1}{4} \)
(ii) \( \because \) There are no even sums
\( \therefore \) Possible outcomes = 0
\( \Rightarrow P_{\text{(Sum = an even)}} = \frac{0}{12} = 0 \)
(iii) \( \because \)
\( 6 + 3 = 9 \)
\( 2 + 7 = 9 \)
\( 4 + 7 = 11 \)
\( 4 + 5 = 9 \)
\( 6 + 5 = 11 \)
\( 6 + 7 = 13 \)
There are six sums which are more than 7
\( \therefore \) Possible outcomes = 6
\( \Rightarrow P_{\text{(Sum more than 7)}} = \frac{6}{12} = \frac{1}{2} \)
Question. A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result, i.e., three heads or three tails, and loses otherwise, calculate the probability that Hanif will lose the game.
Answer: For Solution, please see the solution of Q. 23 of the Textbook Exercise 15.1.
Question. Find the probability that a number selected at random from numbers 3, 4, 5, ....., 25 is prime.
Answer: Total numbers are 23.
\( \therefore \) Number of possible outcomes = 23
Since, prime numbers are 3, 5, 7, 11, 13, 17, 19 and 23.
\( \therefore \) Number of favourable outcomes = 8
\( \Rightarrow P(E) = \frac{8}{23} \).
Question. The king, queen and jack of diamonds are removed from a pack of 52 cards are then the pack is well-shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of
(i) diamonds
(ii) a Jack
Answer: \( \because \) There are 52 card in the pack.
And number of cards removed = 3 [1 king + 1 queen + 1 jack = 3 cards]
\( \therefore \) Remaining cards = 52 \( - \) 3 = 49
\( \therefore \) (i) \( P_{\text{(a diamond)}} = \frac{13 - 3}{49} = \frac{10}{49} \) [\( \because \) Total diamonds are 13]
(ii) \( P_{\text{(a jack)}} = \frac{4 - 1}{49} = \frac{3}{49} \) [\( \because \) Total jacks are 4]
Question. A bag contains 5 red, 4 blue and 3 green balls. A ball is taken out of the bag at random. Find the probability that the selected ball is
(i) of red colour
(ii) not of green colour.
Answer: Total number of balls = 5 + 4 + 3 = 12
\( \Rightarrow \) Number of possible outcomes = 12
(i) \( \because \) Number of red balls = 5
\( \therefore \) Favourable outcomes = 5
\( \Rightarrow P_{\text{(red ball)}} = \frac{5}{12} \)
(ii) \( \because \) Number of green balls = 3
\( \therefore \) Number of ball which are not green = 12 \( - \) 3 = 9
\( \Rightarrow \) Favourable outcomes = 9
\( \therefore P_{\text{(not green)}} = \frac{9}{12} = \frac{3}{4} \).
Question. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing a
(i) face card
(ii) card which is neither a king nor a red card.
Answer: Total number of cards = 52
(i) Total number of face cards = 12 [4 Jacks + 4 Queens + 4 Kings]
\( \therefore \) Number of favourable outcomes = 12
\( \Rightarrow P_{\text{(face)}} = \frac{12}{52} = \frac{3}{13} \)
(ii) Number of kings = 4
Number of red cards = 13 + 13 = 26
\( \therefore \) Number of cards that are neither a red nor a king = 52 \( - \) 4 \( - \) 26 + 2 (red kings)
= 24
\( \Rightarrow \) Favourable outcomes = 24
\( \therefore P_{\text{(neither king nor red)}} = \frac{24}{52} = \frac{6}{13} \).
Question. A bag contains tickets, numbered 11, 12, 13, ....., 30. A ticket is taken out from the bag at random.
Find the probability that the number on the drawn ticket
(i) is a multiple of 7
(ii) is greater than 15 and a multiple of 5.
Answer: Total number of tickets = 20 [\( \because \) Numbers from 11 to 30 are 20]
(i) \( \because \) Multiples of 7 are 14, 21 and 28
\( \therefore \) Number of favourable outcomes = 3
\( \Rightarrow P_{\text{(a multiple of 7)}} = \frac{3}{20} \)
(ii) \( \because \) The numbers that are greater than 15 and multiples of 5 are: 20, 25 and 30
\( \therefore \) Number of favourable outcomes = 3
\( \Rightarrow P_{\text{(multiples of 5 and greater than 15)}} = \frac{3}{20} \).
Question. A bag contains 4 red, 5 black and 3 yellow balls. A ball is taken out of the bag at random. Find that the ball taken out is of:
(i) yellow colour
(ii) not of red colour.
Answer: Total number of balls = 4 + 5 + 3 = 12
\( \Rightarrow \) Total number of possible outcomes = 12
(i) \( \because \) Number of yellow balls = 3
\( \therefore \) Number of favourable outcomes = 3
\( \Rightarrow P_{\text{(yellow)}} = \frac{3}{12} = \frac{1}{4} \)
(ii) Number of balls that are not red = 12 \( - \) 4 = 8 [\( \because \) There are 4 red balls]
\( \therefore \) Favourable outcomes = 8
\( \Rightarrow P_{\text{(not red)}} = \frac{8}{12} = \frac{2}{3} \).
Question. A coin is tossed two times. Find the probability of getting at most one head.
Answer: Since, the coin is thrown two times.
\( \therefore \) Possible out comes = 4
Favourable outcomes are TT, TH, HT
i.e., Number of favourable outcomes = 3
\( \therefore P \text{(atmost one head)} = \frac{3}{4} \).
Question. There are 40 students in class X of whom 25 are girls and 15 are boys. The class teacher has to select one student as a class representative. She writes the name of each student on a separate card.
The cards being identical and she puts cards in a bag and stirs throughly. She then draws one card from the bag. What is the probability that the name written on the card is the name of a:
(i) girl (ii) a boy
Answer: Total number of students = 40
\( \Rightarrow \) Number of possible outcomes = 40
(i) \( \because \) There are 25 girls in the class
\( \therefore \) Number of favourable outcomes = 25
\( \Rightarrow P_{\text{(name of a girl)}} = \frac{25}{40} = \frac{5}{8} \)
(ii) \( \because \) Number of boys = 15
\( \therefore \) Number of favourable outcomes = 15
\( \Rightarrow P_{\text{(name of a boy)}} = \frac{15}{40} = \frac{3}{8} \)
Question. Cards, marked with numbers 5 to 50, are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is:
(i) a prime number less than 10.
(ii) a number which is a perfect square.
Answer: Numbers from 5 to 50 are 46.
\( \therefore \) Total number of possible outcomes = 46.
(i) Prime numbers (less than 10) are 5, 7.
\( \therefore \) Favourable outcomes = 2
\( \Rightarrow P_{\text{(prime number less than 10)}} = \frac{2}{46} = \frac{1}{23} \)
(ii) Perfect square are 9, 16, 25, 36 and 49
\( \therefore \) Number of favourable outcomes = 5
\( \Rightarrow P_{\text{(perfect square)}} = \frac{5}{46} \).
Question. A die is thrown once. Find the probability of getting:
(i) an even prime number.
(ii) a multiple of 3.
Answer: Total numbers on the faces of a die are 1, 2, 3, 4, 5 and 6
\( \Rightarrow \) Number of favourable outcomes = 6
(i) Even prime number is only one i.e. 2
\( \therefore \) Favourable outcome = 1
\( \Rightarrow P_{\text{(even prime number)}} = \frac{1}{6} \)
(ii) Multiples of 3 are 3 and 6
\( \therefore \) Favourable outcomes are 2.
\( \Rightarrow P_{\text{(multiple of 3)}} = \frac{2}{6} = \frac{1}{3} \).
Question. A die is thrown once. Find the probability of getting:
(i) a prime number
(ii) a number divisible by 2.
Answer: \( \because \) The numbers on the faces of a die are 1, 2, 3, 4, 5 and 6.
\( \therefore \) Number of possible outcomes = 6
(i) Prime numbers are 2, 3 and 5
\( \therefore \) Number of prime numbers = 3
\( \Rightarrow \) Number of favourable outcomes = 3
\( \therefore P_{\text{(prime number)}} = \frac{3}{6} = \frac{1}{2} \)
(ii) Numbers divisible by 2 are 2, 4 and 6
\( \therefore \) Favourable outcomes are 3.
\( \Rightarrow P_{\text{(divisible by 2)}} = \frac{3}{6} = \frac{1}{2} \).
Question. Two dice are thrown simultaneously. What is the probability that
(i) 5 will not come up on either of them?
(ii) 5 will come up on at least one?
(iii) 5 will come up at both dice?
Answer: \( \because \) The two dice are thrown simultaneously
\( \therefore \) Possible outcomes are = 6 \( \times \) 6 = 36
(i) When 5 will not come up on either of them:
Favourable outcomes are: 36 \( - \) 11 = 25
\( \therefore P_{\text{(5 will not come up on either dice)}} = \frac{25}{36} \)
(ii) When 5 will come on at least one dice:
Favourable outcomes are: 36 \( - \) 25 = 11
\( \therefore P_{\text{(5 will come on at least one dice)}} = \frac{11}{36} \)
(iii) When 5 will come up on either dice:
Favourable outcome is only one i.e. (5, 5)
\( \therefore P_{\text{(5 on both dice)}} = \frac{1}{36} \).
Question. Two different dice are rolled simultaneously. Find the probability that the sum of numbers
appearing on the two dice is 10.
Answer: When two different dice are rolled then possible outcomes are :
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
\( \therefore \) Number of total outcomes = 36
\( \because \) Sum of (5, 5), (6, 4) and (4, 6) is 10.
\( \therefore \) No of favourable outcomes = 3
\( \Rightarrow \) Required Probability = \( \frac{3}{36} \) or \( \frac{1}{12} \).
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CBSE Class 10 Mathematics Chapter 14 Probability Study Material
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