At each turn, a gambler bets a certain amount, wins it with probability p and loses it with probability q = 1-p. when he begins playing, he has $k for some integer k > 0 and his goal is to reach $n for some integer n > k, after which he stops playing. he also stops playing if he loses all his money. the gambler has the option of betting $1 at each turn and the option of betting $0.5 at each turn. show that betting $0.5 is a better option if p > 1/2 and that betting $1 is a better option if p < 1/2.

Question

Mathematics

Gray

11 July, 21:56

At each turn, a gambler bets a certain amount, wins it with probability p and loses it with probability q = 1-p. when he begins playing, he has $k for some integer k > 0 and his goal is to reach $n for some integer n > k, after which he stops playing. he also stops playing if he loses all his money. the gambler has the option of betting $1 at each turn and the option of betting $0.5 at each turn. show that betting $0.5 is a better option if p > 1/2 and that betting $1 is a better option if p < 1/2.

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Ruth Fischer · Answer 1

Here two cases of the probability,

p > 1/2 or p <1/2

Here, q = 1 - p

So, When p > 1/2 the q < 1/2 &

When p 1/2

So, in first case (p > 1/2 the q < 1/2) Winning chances is more compare to Loosing, So put the Bet maximum and is of $1

Another case (p 1/2) Winning chances is less compare to Loosing, So put the Bet minimum and is of $0.5