probability and statistics

Question

Find the probability of exactly two successes in five trials of a binomial experiment in which the probability of success is 50%. Round to the nearest tenth of a percent.

Answer 1

This is the same as the classic "coin toss" problems.

If you toss a coin 5 times, there are 2^5 = 32 possible outcomes. To answer any specific question, we need to account for the permutations that are all the same result. Doing this, we recognize that there are 5 ways to get exactly 1 head, so the probability is 5/32.

How to get exactly 2 heads?

1 = heads

1 1 0 0 0

1 0 1 0 0

1 0 0 1 0

1 0 0 0 1

0 1 1 0 0

0 1 0 1 0

0 1 0 0 1

0 0 1 1 0

0 0 1 0 1

0 0 0 1 1

total 10, probability = 10/32 = 5/16

exactly 3 heads---same as exactly 2**

exactly 4 heads---same as exactly 1**

finally, the odds of exactly 5 heads, same as exactly 0, 1/32

Add them all up:

1/32 + 5/32 + 10/32 + 10/32 + 5/32 + 1/32 = 32/32

**exactly 3 heads means exactly 2 tails, which is the same odds as exactly 2 heads.....similar logic for exactly 4 heads.

Answer 2

₅C₂(p)²(q)^(5-2) Where Combinations, p = probability of success, q = probability of failure

₅C₂(.5)²(.5)³ = .3125 or .3