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 right, and your friend is NOT psychic, the the probability of guessing correctly on each flip is .5. Your friend correctly guesses 15 out of 20 flips.

What is the probability of your friend correctly guessing exactly 15 out of 20 flips if he is NOT really psychic?

To calculate the probability of your friend correctly guessing exactly 15 out of 20 coin flips if he is NOT really psychic, we need to use the binomial probability formula.

The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
where:
- P(x) represents the probability of getting exactly x successes
- n is the total number of trials (coin flips, in this case)
- x is the number of successful outcomes (correct guesses, in this case)
- p is the probability of success on a single trial (probability of guessing correctly on each flip)

In our case:
- n = 20 (total number of coin flips)
- x = 15 (number of correct guesses)
- p = 0.5 (probability of guessing correctly on each flip)

Using the binomial probability formula, we can substitute these values and calculate the probability of your friend correctly guessing exactly 15 out of 20 flips:

P(15) = (20C15) * (0.5^15) * ((1-0.5)^(20-15))

Now, let's calculate it step by step:

1. Calculate the number of combinations (20C15):
(20C15) = 20! / (15! * (20-15)!)
= 20! / (15! * 5!)

2. Calculate the probability of getting 15 heads on 20 flips:
(0.5^15) = 0.5^15 = 0.000030517578125

3. Calculate the probability of getting 5 tails on 20 flips:
(1-0.5)^(20-15) = 0.5^5 = 0.03125

4. Multiply all the calculated values together:
P(15) = (20C15) * (0.5^15) * ((1-0.5)^(20-15))
= (20! / (15! * 5!)) * 0.000030517578125 * 0.03125

After performing the calculations, we can find that the probability of your friend correctly guessing exactly 15 out of 20 flips if he is NOT really psychic is approximately 0.00145721435546875 or 0.14%.

.015