Answer
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.
