In a game of Kikuyu a player wins a game only by obtaining two heads when he/she tosses two fair coins. a) find the probability that more than 6 attempts are needed to win a game. b) what is the smallest value of n if there is to be at least a 0.9 probability of winning a game on or befofore the nth attempt
Answers
Given : In a game of Kikuyu a player wins a game only by obtaining two heads when he/she tosses two fair coins
To find : a . probability that more than 6 attempts are needed to win a game
b. smallest value of n if there is to be at least a 0.9 probability of winning a game on or before the nth attempt
Solution:
Probability of getting both heads = (1/2)(1/2) = 1/4
Probability of not getting 2 Heads =1 - 1/4 = 3/4
More than 6 attempts needed to win a game
Probability in winning nth attempt = (3/4)ⁿ⁻¹(1/4)
7 attempts or more attempts to win a game
= (3/4)⁶(1/4) + (3/4)⁷(1/4) + (3/4)⁸(1/4) +........................
a = (3/4)⁶(1/4)
r = 3/4
S = a/(1 - r)
= (3/4)⁶(1/4) /(1 - 3/4)
= (3/4)⁶
(3/4)⁶ is the probability that more than 6 attempts are needed to win a game
Probability in winning nth attempt = (1/4)(3/4)ⁿ⁻¹
probability of winning a game on or before the nth attempt
(1/4) + (3/4)(1/4) + (3/4)²(1/4) + ............................+ (3/4)ⁿ⁻¹(1/4)
a = 1/4 r = 3/4
Sₙ = a(1 - rⁿ)/(1 - r)
= (1/4) (1 - (3/4)ⁿ)/(1 - 3/4)
= 1 - (3/4)ⁿ
1 - (3/4)ⁿ ≥ 0.9
=> (3/4)ⁿ ≤ 0.1
=> n = 9
9 attempt required
smallest value of n = 9 if there is to be at least a 0.9 probability of winning a game on or befofore the nth attempt
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