Binomial distribution table

Binomial distribution cumulative probability table P(X ≤ k) for n = 5, 10, 15, 20 and common values of p. Interactive calculator included.

n = 5

k  p 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0 0.77378 0.59049 0.44371 0.32768 0.23730 0.16807 0.11603 0.07776 0.05033 0.03125
1 0.97741 0.91854 0.83521 0.73728 0.63281 0.52822 0.42841 0.33696 0.25622 0.18750
2 0.99884 0.99144 0.97339 0.94208 0.89648 0.83692 0.76483 0.68256 0.59313 0.50000
3 0.99997 0.99954 0.99777 0.99328 0.98438 0.96922 0.94598 0.91296 0.86878 0.81250
4 1.00000 0.99999 0.99992 0.99968 0.99902 0.99757 0.99475 0.98976 0.98155 0.96875
5 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

What is the binomial distribution?

The binomial distribution models the number of successes \(k\) in \(n\) independent Bernoulli trials, each with success probability \(p\). The probability mass function is:

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

How to use this table

The table gives \(P(X \leq k \mid n, p)\) — the cumulative probability of at most \(k\) successes.

  1. Select \(n\) (number of trials) using the table selector above
  2. Find row \(k\) and column \(p\)
  3. Read \(P(X \leq k)\) from the cell

Useful identities: - \(P(X = k) = P(X \leq k) - P(X \leq k-1)\) - \(P(X > k) = 1 - P(X \leq k)\) - \(P(X \geq k) = 1 - P(X \leq k-1)\)

Worked example

A fair coin is flipped 10 times (\(n = 10\), \(p = 0.50\)). What is the probability of getting at most 3 heads?

Select n = 10, row k = 3, column p = 0.50 → \(P(X \leq 3) = 0.17188\).