You have a friend who claims to 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 of tails. If you are right, and your friend is NOT psychic, then the probability of guessing correctly on each flip is .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?

prob of getting 15 of 20 correct

= C(20,15) (1/2)^20 = 15504/1048576 = .014785..
So what is your conclusion?

To calculate the probability of your friend correctly guessing 15 or more out of 20 flips if he is NOT really psychic, we can use the binomial probability formula.

The binomial probability formula is given by P(X=x) = C(n, x) * p^x * (1-p)^(n-x), where:
- P(X=x) is the probability of x successes in n trials
- C(n, x) is the binomial coefficient, which represents the number of ways to choose x successes out of n trials
- p is the probability of success on a single trial
- (1-p) is the probability of failure on a single trial
- n is the total number of trials

In this case, the probability of success (correct guess) on a single flip is 0.5 (since a fair coin has an equal probability of landing heads or tails), and the total number of trials (flips) is 20.

To find the probability of your friend correctly guessing 15 or more flips, we need to calculate the probabilities of 15, 16, 17, 18, 19, and 20 successes and sum them up.

P(X >= 15) = P(X=15) + P(X=16) + P(X=17) + P(X=18) + P(X=19) + P(X=20)

Using the binomial probability formula, we can calculate each of these probabilities:

P(X=15) = C(20,15) * 0.5^15 * (1-0.5)^(20-15)
P(X=16) = C(20,16) * 0.5^16 * (1-0.5)^(20-16)
P(X=17) = C(20,17) * 0.5^17 * (1-0.5)^(20-17)
P(X=18) = C(20,18) * 0.5^18 * (1-0.5)^(20-18)
P(X=19) = C(20,19) * 0.5^19 * (1-0.5)^(20-19)
P(X=20) = C(20,20) * 0.5^20 * (1-0.5)^(20-20)

Calculating these probabilities, we find:

P(X=15) ≈ 0.01487
P(X=16) ≈ 0.00328
P(X=17) ≈ 0.00042
P(X=18) ≈ 0.00003
P(X=19) ≈ 0.000001
P(X=20) ≈ 0.00000003

Summing up these probabilities, we get:

P(X >= 15) ≈ 0.01871

Therefore, if your friend is NOT really psychic, the probability of correctly guessing 15 or more out of 20 flips would be approximately 0.01871 or 1.871%.

To calculate the probability of your friend correctly guessing 15 or more out of 20 flips if he is not really psychic, we can use the binomial probability formula.

The binomial probability formula is:
P(x) = nCx * p^x * (1-p)^(n-x)

Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successes
p is the probability of success in a single trial
(1-p) is the probability of failure in a single trial
nCx is the binomial coefficient, which can be calculated as n! / (x! * (n-x)!)

In this case, we want to calculate the probability of getting 15, 16, 17, 18, 19, or 20 correct guesses out of 20 flips. Since your friend has a 50% chance of correctly guessing each flip, p = 0.5. The total number of trials is 20, so n = 20.

Let's calculate the probability using the formula for each number of correct guesses and then sum them up:

P(15 or more) = P(15) + P(16) + P(17) + P(18) + P(19) + P(20)

P(15) = 20C15 * 0.5^15 * (1-0.5)^(20-15)
P(16) = 20C16 * 0.5^16 * (1-0.5)^(20-16)
P(17) = 20C17 * 0.5^17 * (1-0.5)^(20-17)
P(18) = 20C18 * 0.5^18 * (1-0.5)^(20-18)
P(19) = 20C19 * 0.5^19 * (1-0.5)^(20-19)
P(20) = 20C20 * 0.5^20 * (1-0.5)^(20-20)

Calculating each of these probabilities using the values above (nCx can be calculated using the factorial function):

P(15) = 20C15 * 0.5^15 * 0.5^5
P(16) = 20C16 * 0.5^16 * 0.5^4
P(17) = 20C17 * 0.5^17 * 0.5^3
P(18) = 20C18 * 0.5^18 * 0.5^2
P(19) = 20C19 * 0.5^19 * 0.5^1
P(20) = 20C20 * 0.5^20 * 0.5^0

Finally, compute the sum:

P(15 or more) = P(15) + P(16) + P(17) + P(18) + P(19) + P(20)

By calculating these probabilities using the binomial probability formula, you will be able to find the probability of your friend correctly guessing 15 or more out of 20 flips if he is not really psychic.