The Binomial Distribution Explained Step by Step

The binomial distribution gives the probability of exactly $k$ successes in $n$ independent trials, each with the same probability of success $p$. The formula is $\binom{n}{k}p^k(1-p)^{n-k}$: the number of ways to choose which trials succeed, times the probability of each such way.

The four conditions (BINS)

• Binary: each trial is success or failure.
• Independent trials.
• N: a fixed number of trials.
• Same probability $p$ each time.

The formula

Worked example 1: coin flips

Exactly 6 heads in 10 fair flips:

Worked example 2: at least 3 sixes

Roll 12 dice. The chance of at least three sixes uses the complement:

Mean and standard deviation

For 12 dice and sixes: $\mu = 2$, $\sigma \approx 1.29$.

Common mistakes

• Forgetting the $\binom{n}{k}$ factor.
• Using the binomial when the trials are not independent (drawing without replacement from a small group).
• Confusing "at least 3" with "more than 3".

Practice questions

1. Probability of exactly 2 heads in 4 fair flips?
2. Mean number of heads in 50 flips?
3. Probability of no sixes in 3 rolls?

Answers: 1) $\frac{6}{16} = 0.375$ 2) 25 3) $\left(\frac{5}{6}\right)^3 \approx 0.5787$

The binomial distribution calculator charts the whole distribution with your answer highlighted. Counting comes from the permutation and combination calculator, explained in permutations vs combinations.

Frequently asked questions

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