Question

Two players share an unbiased coin and take it in turns to play "the race". On player 1's turn, he tosses the coin once: if it comes up heads, he scores one point; if it comes up tails, he scores nothing. On player 2's turn, she chooses a positive integer t and tosses the coin t times: if it comes up all heads, she scores 2t-1 points; otherwise, she scores nothing. Player 1 goes first. The winner is the first to 100 or more points. On each turn player 2 selects the number, t, of coin tosses that maximises the probability of her winning. What is the probability that player 2 wins?

Answer 1

Answer:

for t = 2 she has maximum probability

Step-by-step explanation:

Player 1  getting head probability = 1/2

to Score 100 points he probably needs 200 turns

and min 100 turns

Player 2 select number t

Probability that all head turned up = (1/2)^t

Points she get = 2t-1

let say t = 1

probability of turning head = 1/2

Point she get = 2*1 - 1 = 1

to score 100 point probably she need 200 turns

she turns second so she has lesser chance to win

let say t = 2

probability of turning head = (1/2)² = 1/4

Point she get = 2*2 - 1 = 3

to score 100 or more point probably she need 136 turns

She has more chances to win

let say t = 3

probability of turning head = (1/2)³= 1/8

Point she get = 2*3 - 1 = 5

to score 100 or more point probably she need 160 turns

She has more chances to win

let say t = 4

probability of turning head = (1/2)⁴= 1/16

Point she get = 2*4 - 1 = 7

to score 100 or more point probably she need 240 turns

She has less chances to win

for t = 2 she has maximum probability