You have a friend who claims to be psychic. You don't believe this, so you test your friend by flipping a coin 20 times and having him predict whether each flip is heads or tails. If you are correct and your friend is not psychic, then the probability of guessing correctly on each flip is 0.5. Your friend correctly guesses 15 out of the 20 flips. What is the probability of your friend correctly guessing 15 or more out of 20 flips if he is not really psychic? (Give your answer to at least three decimal places.)

The probability of your friend correctly guessing 15 or more out of 20 flips if he is NOT really psychic can be calculated using binomial probability. In this case, the probability of guessing correctly on each flip is 0.5. We can use the binomial probability formula to calculate the probability of getting 15, 16, 17, 18, 19, or all 20 correct guesses. The formula is:[tex]P(X = k) = C(n, k) * p^k * q^{(n-k)[/tex], where n is the number of trials (20), k is the number of successful outcomes (15, 16, 17, 18, 19, or 20), p is the probability of success (0.5), q is the probability of failure (1-0.5=0.5), and C(n, k) is the number of ways to choose k successful outcomes from n trials (n choose k). We can then sum up the individual probabilities for each possible number of successful outcomes to get the final probability.

Using this formula, we can find the probability of your friend correctly guessing 15 or more out of 20 flips if he is NOT really psychic to be approximately 0.058.

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mtosi17 mtosi17 · Answer 2 · 2024-11-20T05:59:15-05:00

Answer:

Given that your friend in not really psychic, he has a probability of P=1.5% of guessing right 15 out of 20 coin flips.

Step-by-step explanation:

We have a binomial distribution problem.

We have to calculate the probability of correctly guessing 15 out of 20 flips of a coin (probability of success for every trial: p=0.5).

We can calculate that with the binomial distribution formula:

[tex]P(x)=\frac{n!}{x!(n-x)!}p^x(1-p)^{n-x} \\\\P(15)=\frac{20!}{15!*5!}0.5^{15}*0.5^5\\\\P(15)=15504*0.5^{20}=15504* 0.00000095367 = 0.015[/tex]

Then we can conclude that, given that your friend in not really psychic, it has only 1.5% of guessing right 15 out of 20 coin flips.