Problems on Not Mutually Exclusive Independent Events

Type (1) (c) Not mutually exclusive, independent events

(i) P(AUB) = P(A) + P(B) - P(A∩B)

(ii) P(A∩B) = P(A) . P(B)

Example 1.1.16

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, what is the probability of getting 2 tails and 1 head.

Solution:

The sample space S = {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}

Since a coin is biased and a head is twice as likely to occur as a tail, P(H) = 2/3, and P(T) = 1/3

Let A be the event of getting 2 tails and 1 head in the 3 tosses of the coin.

Then A = {TTH, THT, HTT}

The outcomes of the 3 tosses are independent,

P(TTH) = P(T∩T∩H) = P(T) P(T) P(H) = (1/3)(1/3)(2/3) = 2/27

|||ly P(THT) = 2/27; P(HTT) = 2/27

Hence P(A) = (2/27) + (2/27) + (2/27) = 6/27 = 2/9

Example 1.1.17

A can hit a target in 4 out of 5 shots and B can hit the target in 3 out of 4 shots. Find the probability that (i) the target being hit when both try. (ii) the target being hit by exactly one person.

Solution :

Let A, B the events

A hit the target P(A) = 4/5

B hit the target P(B) = 3/4

(i) The events A and B are not mutually exclusive because both of them hit the target.

(ii) The target being hit by exactly one person bas

Example 1.1.18

A is known to hit the target in 2 out of 5 shots whereas B is known to hit the target in 3 out of 4 shots. Find the probability of the target being hit when they both try?

Solution :

A -> be the event that 'A' hits the target

B -> be the event that 'B' hits the target

Given : P(A) = 2/5; P(B) = 3/4

To find P(AUB)

Example 1.1.19

One card is drawn from a deck of 52 cards. What is the probability of the card being either red or a king.

Solution:

Let A = {an event that the card drawn is red}

B = {an event that the card drawn is king}

AUB = {an event that a card to be either red or a king}

There are two red coloured king cards.

Example 1.1.20

A total of 36 members of a club play tennis, 28 play squash, and 18 play badminton. Furthermore, 22 of the members play both tennis and squash 12 play both tennis and badminton, 9 play both squash and badminton, and 4 play all the three sports. How many members of this club play atleast one of these sports?

Solution :

Let N -> the number of members of the club

C -> any subset

P(C) = number of members in C / N

Now T -> set of members that plays tennis

S -> set of members that plays squash

B -> set of members that plays badminton

Hence we can conclude that 43 members play atleast one of the sports.

Example 1.1.21

If A and B are events with P(A) = 3/8, P(B) = 1/2 and P(A∩B) = 1/4, find P(A^C∩B^C) [A.U N/D 2006]

Solution: 

We know that

Example 1.1.22

If A and B are independent events with P(A) = 0.4 and P(B) = 0.5 find P(A U B). [A.U. Dec, 96]

Solution

Example 1.1.23

Let events A and B be independent with P(A) = 0.5 and P(B) = 0.8. Find the probability that neither of the events A nor B occurs. [A.U. May 2000]

Solution :

Example 1.1.24

Event A and B are such that P(A + B) = 3/4, P(AB) = 1/4 and P(Ā) = 2/3 find P(B). [A.U N/D 2004]

Solution :

By the addition theorem on probability we have,

Example 1.1.25

If P(A) = 0.4, P(B) = 0.7 and P(A∩B) = 0.3 find the probability that neither A nor B occurs.

Solution :

Example 1.1.26

If you twice a flip a balanced coin, what is the probability of getting atleast one head ?

Solution:

When we flip a balanced coin, the sample space will be,

S = {HH, HT, TH, TT}

Example 1.1.27

Prove that for any event A in S, P(A∩Ā) = 0

Solution : 

We know that P(S) = 1 and A ∩ Ā = S

EXERCISES 1.1

1. State the axioms of probability

2. Define mutually exclusive events with an example.

3. Out of 50 students in a class, what is the probability of a single student to opt for a picnic. [Ans. 0.02]

4. What is the probability of obtained two heads in two throws of a single coin? [Ans. 1/4]

5. What is the probability of picking an ace and a king from a deck of 52 cards?[Ans. 8/663]

6. From a bag containing 3 red and 2 black balls, 2 balls are drawn at random. Find the probability that they are of the same colour. [Ans. 2/5]

7. Prove that the probability of an impossible event is zero.

8. When A and B are 2 mutually exclusive events such that P(A) = 1/2 and P(B) = 1/3,

find P(AUB) and P(A∩B). [Ans. P(AUB) = 5/6, P(A∩B) = 0]

9. A fair coin is tossed 5 times what is the probability of having atleast one head ? [Ans. 31/32]

10. A card is drawn at random from a well shuffled pack, what is the probability that it is a heart or a queen. [Ans. 2/13]

11. Given that P(A) = 0.31, P(B) = 0.47 A and B are mutually exclusive. Then find [Ans. 0.31]

12. If P(A) = 0.35, P(B) = 0.73 and P(A∩B) = 0.14 find  [Ans. 0.86]

13. A card is drawn from a well shuffled pack of 52 cards. What is the probability that it is either clever or king. [Ans. 4/13]

14. If BCA, prove that 

15. Given P(A) = 1/3, P(B) = 1/4 P(A∩B) = 1/6, find the following probability 

16. If A and B are two independent events then 

17. It is given that P(A U B) = 5/8, P(A∩B) = 1/3 and  = 1/2. Show that the events A and B are independent.

18. Given P(A) = 0.35, P(B) = 0.73 and P(A∩B) = 0.14, find (a)P(AUB) (b)P(A∩B) 

19. Given P(A) = 0.3, P(B) = 0.5 and P(A∩B) = 0.24 find