Binomial Probability Calculator

Answer:

$ P(5) $ Probability of exactly 5 successes: 0.29848476328125

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Solution:

$P(5)$ Probability of exactly 5 successes

If using a calculator, you can enter $ \text{trials} = 7 $, $ p = 0.65 $, and $ X = 5 $ 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 = 5 $. $$ P(5) = \frac{7!}{5!(7-5)!} \cdot 0.65^5 \cdot (1-0.65)^{7-5} $$ Evaluating the expression, we have $$ P(5) = 0.29848476328125 $$

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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