Bernoulli Trials Probability Calculator

Calculator
Calculate the probability \(P(A)\), where \(A\) \(=\) "there will exactly \(s\) successes in \(t\) trials".
Number of trials \(t\)
Number of successes \(s\)
Probability of success \(p\)

Imagine some experiment (for example, tossing a coin) that only has two possible outcomes. Such an experiment is called Bernoulli trial.

Now image a series of such experiments. What is the probability that tossing a fair coin 5 times we will get exactly 2 heads (and hence 3 tails)?

Let \(A\)— "There will be 2 heads in 5 trials". Then, the probability is given by:

\( P(A) = \binom{5}{2} {1\over2}^2 {1 \over 2}^3 = 10 \times {1 \over 32} = 5/16 = 0.3125\)

Generally:

\( P(A) = \binom{n}{k} {p}^{n} {q}^{n-k} \)

Where
\(n\) is the number of trials;
\(k\) is the number of successes;
\(p\) the probability for a success;
\(q\) the probability for a failure;
and \(\binom{p}{q}\) is the binomial coefficient.