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QUESTION: 1

A bag contains 2 yellow, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

Solution:

Total number of balls = 2 + 3 + 2 = 7

► Let S be the sample space.

**n(S)**= Total number of ways of drawing 2 balls out of 7 =^{7}C_{2}

► Let E = Event of drawing 2 balls, none of them is blue.

**n(E)**= Number of ways of drawing 2 balls from the total 5 (= 7-2) balls =^{5}C_{2}

(∵ There are two blue balls in the total 7 balls. Total number of non-blue balls = 7 - 2 = 5)

QUESTION: 2

A die is rolled twice. What is the probability of getting a sum equal to 9?

Solution:

► Total number of outcomes possible when a die is rolled = 6 (∵ any one face out of the 6 faces)

- Hence, total number of outcomes possible when a die is rolled twice, n(S) = 6 x 6 = 36

► E = Getting a sum of 9 when the two dice fall = {(3,6), (4,5), (5,4), (6,3)}

- Hence, n(E) = 4

QUESTION: 3

Three coins are tossed. What is the probability of getting at most two tails?

Solution:

► Total number of outcomes possible when a coin is tossed = 2 (∵ Head or Tail)

- Hence, total number of outcomes possible when 3 coins are tossed, n(S) = 2 x 2 x 2 = 8

(∵ S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH})

► E = event of getting at most two Tails = {TTH, THT, HTT, THH, HTH, HHT, HHH}

- Hence, n(E) = 7

QUESTION: 4

When tossing two coins once, what is the probability of heads on both the coins?

Solution:

► Total number of outcomes possible when a coin is tossed = 2 (∵ Head or Tail)

- Hence, total number of outcomes possible when two coins are tossed, n(S) = 2 x 2 = 4

(∵ Here, S = {HH,HT,TH,TT})

► E = event of getting heads on both the coins = {HH}

- Hence, n(E) = 1

QUESTION: 5

What is the probability of getting a number less than 4 when a die is rolled?

Solution:

- Total number of outcomes possible when a die is rolled = 6

(∵ any one face out of the 6 faces) i.e., n(S) = 6 - E = Getting a number less than 4 = {1, 2, 3}

Hence, n(E) = 3

QUESTION: 6

A bag contains 4 black, 5 yellow and 6 green balls. Three balls are drawn at random from the bag. What is the probability that all of them are yellow?

Solution:

Total number of balls = 4 + 5 + 6 = 15

► Let S be the sample space.

**n(S)**= Total number of ways of drawing 3 balls out of 15 =^{15}C_{3}

► Let E = Event of drawing 3 balls, all of them are yellow.

**n(E)**= Number of ways of drawing 3 balls from the total 5 =^{5}C_{3}

(∵ there are 5 yellow balls in the total balls)

[∵ ^{n}C_{r} = ^{n}C_{(}_{n-r)}. So ^{5}C_{3} = ^{5}C_{2}. Applying this for the ease of calculation]

QUESTION: 7

One card is randomly drawn from a pack of 52 cards. What is the probability that the card drawn is a face card(Jack, Queen or King)

Solution:

- Total number of cards, n(S) = 52
- Total number of face cards, n(E) = 12 (4 Jacks, 4 Queens, 4 Kings)

QUESTION: 8

A dice is thrown. What is the probability that the number shown in the dice is divisible by 3?

Solution:

- Total number of outcomes possible when a die is rolled, n(S) = 6 (? 1 or 2 or 3 or 4 or 5 or 6)
- E = Event that the number shown in the dice is divisible by 3 = {3, 6}

Hence, n(E) = 2

QUESTION: 9

John draws a card from a pack of cards. What is the probability that the card drawn is a card of black suit?

Solution:

Total number of cards, n(S) = 52

Total number of black cards, n(E) = 26

QUESTION: 10

There are 15 boys and 10 girls in a class. If three students are selected at random, what is the probability that 1 girl and 2 boys are selected?

Solution:

► Let S be the sample space.

**n(S)**= Total number of ways of selecting 3 students from 25 students =^{25}C_{3}

► Let E = Event of selecting 1 girl and 2 boys

**n(E)**= Number of ways of selecting 1 girl and 2 boys

15 boys and 10 girls are there in a class. We need to select 2 boys from 15 boys and 1 girl from 10 girls

__Number of ways in which this can be done: __

^{15}C_{2} × ^{10}C_{1}

Hence n(E) = ^{15}C_{2} × ^{10}C_{1}

### Probability - 1

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