Counting Successes with the Binomial Model: Quick Revision

A brief note on Counting Successes with the Binomial Model. Read it before a test. Try each question at the end before you open its answer.

When a count is binomial

• Two outcomes on each trial, success or failure.
• Independent trials. Picks from a group are close enough when the sample is under $10\%$ of the group.
• A fixed number $n$ of trials.
• The same chance $p$ on every trial.

The formula

• One order: $p^k(1 - p)^{n-k}$.
• Number of orders: $\binom{n}{k}$, from Pascal's triangle or math.comb(n, k).
• Probability: $P(X = k) = \binom{n}{k}p^k(1 - p)^{n-k}$.

Centre, spread and shape

• Mean: $\mu = np$.
• Standard deviation: $\sigma = \sqrt{np(1 - p)}$.
• Usual range: $np \pm 2\sigma$.
• Proportion: mean $p$, standard deviation $\sqrt{p(1 - p)/n}$.
• Shape: small $p$ skews right, $p = 0.5$ is symmetric, large $p$ skews left. A large $n$ gives a bell.

A worked example

Ten made-up light switches each fail with probability $0.05$, independently.

1. None fail. $0.95^{10} \approx 0.5987$.
2. Exactly one fails. $10 \times 0.05 \times 0.95^{9} \approx 0.3151$.
3. At least two fail. $1 - 0.5987 - 0.3151 \approx 0.0861$.
4. Centre and spread. $\mu = 0.5$ and $\sigma = \sqrt{0.475} \approx 0.6892$ switches.

Practice with new numbers

A program to try

The program checks the worked example.

Mistakes to avoid

• Leaving out $\binom{n}{k}$.
• Using the binomial when $n$ is not fixed or the chance changes.
• Taking the variance for the standard deviation.
• Calling every binomial symmetric.

Check yourself

1. How many orders give exactly $2$ successes in $8$ trials?

2. A count is $B(9, 0.5)$. Find $P(X = 0)$.

3. Find the mean and the standard deviation of $B(36, 0.5)$.

4. Is $B(15, 0.85)$ skewed to the left or to the right?

Sources

• OpenStax, Introductory Statistics, Chapter 4, section on the binomial distribution.