The binomial probability calculator will calculate a probability based on the binomial probability formula.  You will also get a step by step solution to follow.  Enter the trials, probability, successes, and probability type. • Trials, n, must be a whole number greater than 0. This is the number of times the event will occur.
• Probability, p, must be a decimal between 0 and 1 and represents the probability of success on a single trial.
• Successes, X, must be a number less than or equal to the number of trials. This number represents the number of desired positive outcomes for the experiment.
• The probability type can either be a single success (“exactly”), or an accumulation of successes (“less than”, “at most”, “more than”, “at least”).

$P(4)$ Probability of exactly 4 successes: 0.26787094140625

### Solution:

$P(4)$ Probability of exactly 4 successes

If using a calculator, you can enter $\text{trials} = 7$, $p = 0.65$, and $X = 4$ into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula: $$P(X) = \binom{n}{X} \cdot p^X \cdot (1-p)^{n-X}$$ The binomial coefficient, $\binom{n}{X}$ is defined by $$\binom{n}{X} = \frac{n!}{X!(n-X)!}$$ The full binomial probability formula with the binomial coefficient is $$P(X) = \frac{n!}{X!(n-X)!} \cdot p^X \cdot (1-p)^{n-X}$$ where $n$ is the number of trials, $p$ is the probability of success on a single trial, and $X$ is the number of successes. Substituting in values for this problem, $n = 7$, $p = 0.65$, and $X = 4$. $$P(4) = \frac{7!}{4!(7-4)!} \cdot 0.65^4 \cdot (1-0.65)^{7-4}$$ Evaluating the expression, we have $$P(4) = 0.26787094140625$$

### Complete Binomial Distribution Table

If we apply the binomial probability formula, or a calculator's binomial probability distribution (PDF) function, to all possible values of X for 7 trials, we can construct a complete binomial distribution table. The sum of the probabilities in this table will always be 1. The complete binomial distribution table for this problem, with p = 0.65 and 7 trials is:

P(0) = 0.00064339296875
P(1) = 0.00836410859375
P(2) = 0.04660003359375
P(3) = 0.14423819921875
P(4) = 0.26787094140625
P(5) = 0.29848476328125
P(6) = 0.18477628203125
P(7) = 0.04902227890625