### Probability Exercise 02

## Question:

Events A and B are such that P(A)=1/2,P(B)=7/12 and P(not A or not B)=1/4. State whether A and B are independent ?

## Answer:

Step 1: Calculate P(A and B).

P(A and B) = P(A) x P(B) = 1/2 x 7/12 = 7/24

Step 2: Calculate P(A) x P(B) given that A and B are independent.

P(A) x P(B) = 1/2 x 7/12 = 7/24

Step 3: Compare P(A and B) with P(A) x P(B).

Since P(A and B) = P(A) x P(B), it can be concluded that A and B are independent.

## Question:

State true or false. A die marked 1,2,3 in red and 4,5,6 in green is tossed. Let A be the event, ’the number is even’, and B be the event, ’the number is red’. Then A and B are independent events. A True B False

## Answer:

False. A and B are not independent events because the number being even and the number being red are not mutually exclusive events.

## Question:

In a hostel, 60% of the students read Hindi news paper, 40% read English news paper and 20% read both Hindi and English news papers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English news papers. (b) If she reads Hindi news paper, find the probability that she reads English news paper. (c) If she reads English news paper, find the probability that she reads Hindi news paper. A 0.56,0.78,0.76 B 0.20,0.33,0.50 C 0.65,0.45,0.34 D 0.56,0.56,0.65

## Answer:

Answer: D 0.56,0.56,0.65

## Question:

Probability of solving specific independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem. A 0.67,0.78 B 0.66,0.50 C 0.67,0.98 D 0.66,0.44

## Answer:

Answer: D. 0.66, 0.44

## Question:

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

## Answer:

Answer: Step 1: Calculate the total number of possible outcomes from drawing two cards from a pack of 52 playing cards.

There are 52 cards in a pack of playing cards, and since we are drawing two cards without replacement, the total number of possible outcomes is 52 × 51 = 2,652.

Step 2: Calculate the number of outcomes that result in two black cards.

There are 26 black cards in a pack of playing cards, and since we are drawing two cards without replacement, the total number of possible outcomes is 26 × 25 = 650.

Step 3: Calculate the probability of drawing two black cards.

The probability of drawing two black cards is 650/2652, which is equal to 0.246.

## Question:

Two events A and B will be independent, if A A and B are mutually exclusive B P(A′B′)=[1−P(A)][1−P(B)] C P(A)=P(B) D P(A)+P(B)=1

## Answer:

A) False B) False C) True D) True

## Question:

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is A 0 B 1/3 C 1/12 D 1/36

## Answer:

Answer: C 1/12

## Question:

Given two independent events A and B such that P(A)=0.3,P(B)=0.6. Find (i) P(AandB) (ii) P(AandnotB) (iii) P(AorB) (iv) P(neitherAnorB) If the sum of the above probablities is m enter 100m

## Answer:

(i) P(A and B) = 0.3 x 0.6 = 0.18

(ii) P(A and not B) = 0.3 x (1 - 0.6) = 0.12

(iii) P(A or B) = 0.3 + 0.6 - (0.3 x 0.6) = 0.78

(iv) P(neither A nor B) = 1 - (0.3 + 0.6 - (0.3 x 0.6)) = 0.22

100 x (0.18 + 0.12 + 0.78 + 0.22) = 100

## Question:

A die is tossed thrice. Find the probability of getting an odd number at least once.

## Answer:

Step 1: Determine the sample space. The sample space is {1,2,3,4,5,6}

Step 2: Determine the favorable outcomes. The favorable outcomes are {1,3,5}

Step 3: Calculate the probability. The probability of getting an odd number at least once is 3/6 or 1/2.

## Question:

Let E and F be events with P(E)=3/5,P(F)=3/10 and P(E∩F)=1/5. Are E and F independent ?

## Answer:

Step 1: Calculate P(E|F)

P(E|F) = P(E∩F) / P(F)

P(E|F) = 1/5 / 3/10

P(E|F) = 2/3

Step 2: Calculate P(F|E)

P(F|E) = P(E∩F) / P(E)

P(F|E) = 1/5 / 3/5

P(F|E) = 1/3

Step 3: Compare P(E|F) and P(F|E) with P(E) and P(F)

P(E|F) = 2/3 ≠ P(E) = 3/5

P(F|E) = 1/3 ≠ P(F) = 3/10

Step 4: Conclusion

Since P(E|F) and P(F|E) are not equal to P(E) and P(F), E and F are not independent.

## Question:

If A and B are two events such that P(A)=1/4,P(B)=1/2 and P(A∩B)=1/8, find 8P(Aˉ and Bˉ)

## Answer:

Answer: Step 1: P(A) = 1/4

Step 2: P(B) = 1/2

Step 3: P(A∩B) = 1/8

Step 4: P(Aˉ and Bˉ) = P(Aˉ) + P(Bˉ) - P(A∩B)

Step 5: P(Aˉ) = 1 - P(A) = 1 - 1/4 = 3/4

Step 6: P(Bˉ) = 1 - P(B) = 1 - 1/2 = 1/2

Step 7: P(Aˉ and Bˉ) = 3/4 + 1/2 - 1/8 = 7/8

Step 8: 8P(Aˉ and Bˉ) = 8*7/8 = 7

## Question:

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

## Answer:

Answer: Step 1: Calculate the total number of possible combinations of 3 oranges that can be drawn from a box of 15 oranges. This can be calculated using the combination formula, nCr, where n = 15 and r = 3. Therefore, the total number of possible combinations of 3 oranges that can be drawn from a box of 15 oranges is 15C3 = 455.

Step 2: Calculate the total number of favorable outcomes. In this case, the favorable outcome is when all the three oranges drawn are good. The total number of good oranges in the box is 12. Therefore, the total number of favorable outcomes is 12C3 = 220.

Step 3: Calculate the probability of the box being approved for sale. This can be calculated by dividing the total number of favorable outcomes by the total number of possible combinations of 3 oranges. Therefore, the probability of the box being approved for sale is 220/455 = 0.483.

## Question:

If P(A)=3/5 and P(B)=1/5, find 100P(A∩B) if A and B are independent events.

## Answer:

Answer: 0

Explanation: Since A and B are independent events, P(A∩B) = P(A) x P(B)

Therefore, P(A∩B) = 3/5 x 1/5 = 3/25

100P(A∩B) = 100 x 3/25 = 0

## Question:

One card is drawn at random from a well-shuffled deck of 52 cards. In how many of the following cases are the events E and F independent? (i) E: ’the card drawn is a spade’ F: ’the card is drawn is an ace’ (ii) E: ’the card drawn is black’ F: ’the card drawn is a king’ (iii) E: ’the card drawn is a king or queen’ F: ’the card drawn is a queen or jack'.

## Answer:

(i) E and F are not independent because the probability of E and F occurring together (P(E and F)) is not equal to the product of the probabilities of E and F occurring separately (P(E) x P(F)). The probability of E and F occurring together is 1/52, while the probability of E and F occurring separately is 1/4 x 1/13 = 1/52.

(ii) E and F are not independent because the probability of E and F occurring together (P(E and F)) is not equal to the product of the probabilities of E and F occurring separately (P(E) x P(F)). The probability of E and F occurring together is 1/26, while the probability of E and F occurring separately is 1/2 x 1/4 = 1/8.

(iii) E and F are independent because the probability of E and F occurring together (P(E and F)) is equal to the product of the probabilities of E and F occurring separately (P(E) x P(F)). The probability of E and F occurring together is 1/16, while the probability of E and F occurring separately is 1/4 x 1/4 = 1/16.

## Question:

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red. (ii) first ball is black and second is red. (iii) one of them is black and other is red. A 0.56, 0.75, 0.67 B 0.46, 0.56, 0.36 C 0.56, 0.76, 0.76 D 0.19, 0.24, 0.49

## Answer:

Answer: C 0.56, 0.76, 0.76

## Question:

A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not. A True B False

## Answer:

Answer: False

Explanation: A and B are not independent events because the outcome of one event (A) affects the outcome of the other event (B). For example, if the coin lands on heads, then the die must land on 3 in order for both events to be true.

## Question:

Given that the events A and B are such that P(A)=21,P(A∪B)=53 and P(B)=p. Find p if they are (i) mutually exclusive (ii) independent. A 0.5,0.6 B 0.1,0.2 C 0,2,0.4 D 0.1,0.6

## Answer:

Answer: (i) Mutually exclusive: p = 0.1

(ii) Independent: p = 0.2