Question

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he did not have ESP?

Answer 1

The required probability is 11/64 or 0.1719 or 17.2%

Explanation:

Consider the provided information.

A fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct.

Let X be the number of times the man guesses the outcome correctly.

Coin is flipped 10 times.

We want to calculate the probability that he would have done at least this well if he did not have ESP.

Therefore, the probability of guessing 7 or more than it is:

`P(x\geq 7)=P(x=7)+P(x=8)+P(x=9)+P(x=10)`

According to binomial distribution:
`P(x=r)=\binom{n}{r}p^r(1-p)^(n-r)`

Therefore,

`P(x\geq 7)=\binom{10}{7}((1)/(2))^7((1)/(2))^(3) +\binom{10}{8}((1)/(2))^8((1)/(2))^(2) +\binom{10}{9}((1)/(2))^9((1)/(2))^(1) +\binom{10}{10}((1)/(2))^(10)((1)/(2))^(0)`

`P(x\geq 7)=(120)/(1024)+(45)/(1024)+(10)/(1024)+(1)/(1024)\\\\P(x\geq 7)=(11)/(64) \approx 0.1719`

Hence, the required probability is 11/64 or 0.1719 or 17.2%