Question

Consider two binomial experiments. a. The first binomial experiment consists of six trials. How many outcomes have exactly one success, and what are these outcomes? b. The second binomial experiment consists of 20 trials. How many outcomes have exactly 10 successes? exactly 15 successes? exactly 5 successes?

Short answer

For the first experiment, there are 6 outcomes with exactly one success. In the second experiment, there are 184756 outcomes with exactly 10 successes, 15504 outcomes with exactly 15 successes, and 15504 outcomes with exactly 5 successes.

Step-by-step solution

Determine outcomes for the first experiment

For the first experiment, there are six trials and the problem asks to find outcomes having exactly one success. Use the formula for combination \[ C(n,k) = \frac{n!}{k!(n-k)!} \]. When n is 6 and k is 1, it becomes \( C(6,1) = 6 \), which means there are 6 outcomes that represent exactly one success.

Determine outcomes for the second experiment

For the second experiment, again use the combination formula to calculate outcomes with exactly 10, 15 and 5 successes, while there are total 20 trials.For 10 successes, it's \( C(20,10) = 184756 \).For 15 successes, it's \( C(20,15) = 15504 \).And finally, for 5 successes, it's \( C(20,5) = 15504 \). Each calculation uses the formula \[ C(n,k) = \frac{n!}{k!(n-k)!} \] with n as 20 and varying values of k.