Question

Players A and B throw a coin alternately till one of them gets a 'head' and wins the game. Find their respective probabilities of winning.

Answer 1

Answer:

Probability of A winning the game is 2/3

Probability of B winning the game is 1/3

Step-by-step explanation:

Hi,

Probability of getting a head, P(H) = 1/2

Probability of getting a tail = P(T) = 1/2

P(A') represents the probability of complement of event A

For A to win the game, till the time A gets a head, B should be

getting tails

So, A would win  either when A gets a head in his first toss or

he gets a tail in his first toss and B gets tails in his first toss and

then A gets a head in 2nd toss and so on this continuing which

would turn out to be an infinite series

P(A) = P(T) + P(T)P(T)P(T) + P(T)P(T)P(T)P(T)P(H) +..........

= 1/2 + 1/2*1/2*1/2 + 1/2*1/2*1/2*1/2*1/2 +........∞ terms

which is a G.P with first term is 1/2 and common ratio is 1/4

Sum to infinite terms of G.P is given by a/(1 - r) where a is first

term and r is common ratio

So, P(A) = 1/2/[1 - 1/4]

           =  2/3

Hence, Probability of A winning the game is 2/3

Since either A or B should win the game indefinitely,

P(A) + P(B) = 1

so, P(B) = 1 - P(A)

= 1 - 2/3 = 1/3

Hence, Probability of B winning the game is 1/3

Hope, it helps !