Question

Flip 10000 fair coins. You are offered a 1-1 bet that the sum is less than 5000. You can bet 1, 2, ..., 100 dollars. How much will you bet. How much will you bet if someone tells you that the sum of the coins is less than 5100?

Answer 1

Explanation:

Given that 10000 times a fair coin is flipped. Since fair coin prob for head of tail =0.5

If tail comes it will be taken as 1. Hence for the sum to be less than 5000, no of heads <5000

Since trials are large we approximate to normal with mean = np = 5000 and variance = np(1-p) = 2500

Thus no of heads x is normal with (5000, 50)

`P(X<5000) = P(x<5000.5)\\= P(Z<(0.5)/(50) )\\=0.5-0.0040\\=0.4960`

Since this prob is less than 0.5 only minimum amount to be bet i.e. 1 dollar

If sum of coins <5100, then Z =
`(5100-5000)/(50) =2`

P(Z<2) = 0.9500

This time we can bet maximum of 100 dollars as probability is very high nearer to 1.