Answer :

**Given:** 3 persons A, B and C apply for a job of Manager in a Private Company

**To find:** the probability that due to the appointment of C no change take place

**Formula used:**

Bayes’ Theorem:

Given E_{1}, E_{2}, E_{3}....,E_{n} is mutually exclusive and exhaustive events; we can find the conditional probability P(E_{i}|A) for any event A associated with E_{i} as follows:

Let A: Change does not occur

E_{1}: selection of A

E_{2}: selection of B

E_{3}: selection of C

Chances of selection of A, B and C are in the ratio 1:2:4

Probability of selection of A = P(E_{1}) =

Probability of selection of B = P(E_{2}) =

Probability of selection of C = P(E_{3}) =

Probability that A introduces change = 0.8

Probability that A does not introduce change = P(A|E_{1}) = 1 – 0.8 = 0.2

Probability that B introduces change = 0.5

Probability that B does not introduce change = P(A|E_{2}) = 1 – 0.5 = 0.5

Probability that C introduces change = 0.3

Probability that C does not introduce change = P(A|E_{3}) = 1 – 0.3 = 0.7

The probability that due to the appointment of C, there is no change take place, P(E_{3}|A)

Hence, the probability that due to the appointment of C no change take place is **0.7**

**OR**

**Given:** A starts the game of throwing a pair of dice and alternatively by B. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10

**To find:** the probability that B wins

Total outcomes when 2 dice are thrown = 36

{∵ (1, 1); (1, 2); (1, 3) and so on}

Total favorable outcomes for A to get a total of 7 = 7

{(1, 6), (6, 1), (3, 4), (4, 3), (2, 5), (5, 2)}

Hence, Probability of A winning the game = P(A) =

The probability of A losing the game

Total favorable outcomes for B to get a total of 10 = 3

{(4, 6), (6, 4), (5, 5)}

Hence, Probability of B winning the game = P(B) =

The probability of B losing the game

Let X be A wins, Y be A loses, S be B wins, and F be B loses

As A starts the game, the game will stop when B wins

∴ Possible sequence = {YS, YFYS, YFYFYS…………………}

YS denotes A loses then B wins

The probability of B winning in 2^{nd} throw =

The probability of B winning in 4^{th} throw =

This will form a G.P.

Sum of the term of a G.P. upto infinity is given by,

where a is the first term, and r is a common ratio

So,

Hence, the Probability that B wins is

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