When is a count binomial?

What you will be able to do

• check the four conditions of a binomial setting, and spot the one that fails;
• count the orders that give $k$ successes in $n$ trials with a tree and with $\binom{n}{k}$;
• find the probability of one particular order.

Ten seeds in a packet

A made-up seed company says that each of its tomato seeds grows with probability $0.8$. You plant a packet of $10$ seeds and count how many grow. The count $X$ is a random variable. It can be any whole number from $0$ to $10$. Which counts are likely?

Each seed is a trial with two outcomes. We call one of them a success (the seed grows) and the other a failure. The words do not judge. A "success" can be a faulty part, if that is what we count.

The four conditions

A count of successes is binomial when all four of these hold.

1. Two outcomes. Each trial is a success or a failure.
2. Independent trials. One trial does not change the chances of another.
3. A fixed number of trials. The number $n$ is decided before you start.
4. The same chance. Each trial has the same probability of success, $p$.

Some books remember them as BINS: binary, independent, number fixed, same chance. The seed packet passes, if the seeds do not affect each other. Here $n = 10$ and $p = 0.8$.

Here are cases that fail, one condition each.

• Roll a die until a six appears, and count the rolls. The number of trials is not fixed.
• A player takes $20$ shots and grows tired, so later shots score less often. The chance is not the same.
• Pick $3$ students from a class of $12$ without putting anyone back, and count the girls. Each pick changes the class, so the trials are not independent.

When a small group breaks independence

Picking without putting back does change the chances. But if the group is large, the change is tiny. The simulation shows how tiny.

You should find that with $20$ people and a share of $0.4$, the chance drops from $0.4$ to about $0.368$ after one liker is picked. With $200$ people it drops only to about $0.397$. A common rule: if the sample is less than $10\%$ of the group, treat the picks as independent.

Counting the orders

Take $3$ seeds, each growing with probability $p$, and write S for a seed that grows and F for one that does not. The tree has $2 \times 2 \times 2 = 8$ paths.

Exactly two seeds grow on three paths: SSF, SFS and FSS. Each path has two S and one F, so each has probability $p \times p \times (1 - p)$. So $P(X = 2) = 3p^2(1 - p)$.

For $n$ trials, the number of orders with exactly $k$ successes is written $\binom{n}{k}$, read "$n$ choose $k$". It counts the ways to choose which $k$ of the $n$ trials are the successes. For $n = 3$ the numbers are $1$, $3$, $3$ and $1$. They are a row of Pascal's triangle, where each number is the sum of the two above it. Python gives them with math.comb(n, k).

Try it in Python

The program lists every order of $4$ trials, groups the orders by the number of successes, and compares each count with math.comb.

Your turn

The usual mistakes

• Using the binomial when $n$ is not fixed. "Until the first success" is not binomial.
• Forgetting that every order counts. SSF, SFS and FSS are three different results with the same count.
• Treating picks from a small group as independent. Check the $10\%$ rule.
• Thinking "success" means something good. It only names the outcome you count.

Remember these

• Binomial: two outcomes, independent trials, a fixed $n$, the same $p$.
• One order with $k$ successes has probability $p^k(1 - p)^{n - k}$.
• There are $\binom{n}{k}$ such orders, a number from Pascal's triangle.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A made-up shop checks $15$ eggs from a huge delivery, and each egg is cracked with probability $0.02$. It counts the cracked eggs. Is the count binomial?

2. How many orders give exactly $3$ successes in $5$ trials?

3. Each trial succeeds with probability $0.7$. What is the probability of the order S, S, F?

4. A player takes $10$ shots, and her chance of scoring rises as she warms up. Which condition fails?

5. Pascal's triangle has the row $1, 4, 6, 4, 1$. Write the next row.

References

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