A coin is tossed 20 times. A person who claims to have extrasensory perception is asked to predict the outcome of each flip in advance. She predicts correctly on 14 tosses. What is the probability of being correct 14 or more
Times by guessing? Does this probability seem to verify her claim? Use the normal distribution to approximate
The desired probability.
A) 0.0582, no
B) 0.4418, no
C) 0.4418, yes
D) 0.0582, yes

Question

Quizplus · Accepted Answer

The probability of correctly guessing the outcome of a coin toss is 0.5. Using the binomial distribution with n = 20 and p = 0.5, the probability of getting 14 or more correct guesses is:

P(X ≥ 14) = 1 - P(X < 14)
= 1 - ∑(from k=0 to k=13) (20 choose k) (0.5)^20
= 1 - 0.9418
= 0.0582

This probability is quite low, which means that the person's claim of having extrasensory perception is not supported by the evidence. To use the normal distribution to approximate the probability, we can assume that the distribution of the number of correct guesses is approximately normal with mean np = 20(0.5) = 10 and standard deviation sqrt(np(1-p)) = sqrt(5) = 2.236. Then, we can use the standard normal distribution to find:

z = (14 - 10) / 2.236 = 1.79

Using a standard normal table or calculator, we find that the probability of getting a z-score of 1.79 or higher is approximately 0.0379. This is close to, but still lower than, the exact binomial probability of 0.0582. Therefore, the best answer is A) 0.0582, no.

A Coin Is Tossed 20 Times