Binomial Probability

Binomial probability: how to solve

1. Check that the situation is binomial. Use this method when there is a fixed number of trials, each trial has two outcomes, the trials are independent, and the probability of success stays the same each time.

2. Identify the key values. Let:

   • (n) = number of trials
   • (k) = number of successes wanted
   • (p) = probability of success on one trial
   • (1-p) = probability of failure

3. Use the binomial probability formula. The probability of exactly (k) successes is [ P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}. ] First compute the combination (\binom{n}{k}), then raise the probabilities to the correct powers, and multiply.

4. Simplify the final result. Write the answer in exact form if possible. If the exercise gives fractions or decimals that can be reduced, simplify them carefully.

How to check your answer

• Make sure the exponents add up to (n).
• Confirm that the probability is between 0 and 1.
• If you are asked for “at least” or “at most,” add the relevant exact probabilities from the binomial formula.

Common mistake

Do not forget the combination factor (\binom{n}{k}): it counts how many different ways the successes can occur.