Binomial Probability Calculator

Answer:

$ P(3) $ Probability of exactly 3 successes: 0.336415625

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Solution:

$P(3)$ Probability of exactly 3 successes

If using a calculator, you can enter $ \text{trials} = 5 $, $ p = 0.65 $, and $ X = 3 $ 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 = 5 $, $ p = 0.65 $, and $ X = 3 $. $$ P(3) = \frac{5!}{3!(5-3)!} \cdot 0.65^3 \cdot (1-0.65)^{5-3} $$ Evaluating the expression, we have $$ P(3) = 0.336415625 $$

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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 5 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 5 trials is:

P(0) = 0.0052521875
P(1) = 0.0487703125
P(2) = 0.181146875
P(3) = 0.336415625
P(4) = 0.3123859375
P(5) = 0.1160290625