Question

A man claims to have extrasensory perception. As a test, a fair coin is flipped 30 times, and the man is asked to predict the outcome in advance. He gets 24 out of 30 correct. What is the probability that he would have done at least this well if he had no ESP?

Answer 1

0.072%

Step by Step Explanation

We have 30 trials

Flipping of a coin is always independent

Probability of Success = ½

Probability of failure = 1-½ = ½

Since we have a finite number of independent trials with constant probability of success, we use the binomial.

Let X be the number of times the man guesses the outcome correctly (the number of successes). We want to know the probability that he does at least as well as he did in the observed study where he guessed correctly 24 times out of 30. Then,

P(X ≥ 24) = P(X = 24) + P(X = 25) + P(X = 26) +...............+ P(X = 30)

= (30,24) ½^24 * ½^6 + (30,25) ½^25 * ½^5 + (30,26) ½^26 * ½^4 +(30,27) ½^27 * ½^3 + (30,28) ½^28 * ½^2 + (30,29) ½^29 * ½^1 + (30,30) ½^30 * ½^0

= (30,24) ½^30 + (30,25) ½^30 + (30,26) ½^30 +(30,27) ½^30 + (30,28) ½^30 + (30,29) ½^30 + (30,30) ½^30

= ½^30 ( (30,24) + (30,25) + (30,26) +(30,27) + (30,28) + (30,29) + (30,30) )

= ½^30 ( 30!/24!6! + 30!/25!5! + 30!26!4! + 30!27!3! + 30!/28!2! + 30!/29!1! + 30!/30!0!)

= ½^30 (593775 + 142506 + 27405 + 4060 + 435 + 30 + 1)

= ½^30 (768212)

= 192053/268435456

= 0.0007155

If the man did not have ESP, then we would expect him to guess at least 24 out of 30 outcomes correctly approximately 0.072% of the time.