What does "success" mean in this context?

In statistics, "success" does not necessarily mean something good. It simply means the outcome you are tracking. If you are calculating the probability of a machine producing a defective part, a defective part is considered the "success" for the sake of the formula.

How is this different from a normal distribution?

A binomial distribution is discrete, meaning you can only have whole numbers of successes (you cannot flip $2.5$ heads). A normal distribution is continuous and can take on any value, including decimals. However, with a large enough number of trials, a binomial distribution starts to look like a bell-shaped normal distribution.

Can I use the binomial distribution if I am drawing cards from a deck?

Only if you replace the card and shuffle after every draw. If you do not replace the card, the probability of drawing a certain suit changes on the next draw. That violates the rule that the probability of success $p$ must remain constant.

What is a binomial distribution?

A binomial distribution is a statistical model that shows the probability of getting a specific number of "successes" in a set number of repeated, independent trials. Every trial must have exactly two possible outcomes, which we generally call success or failure.

Think of it like flipping a coin 10 times and wanting to know the exact odds of getting exactly 7 heads. You have a fixed number of flips, only two possible outcomes per flip, and the result of one flip does not affect the next. The binomial distribution provides the exact mathematical framework to find that probability.

The Four Conditions (BINS)

To know if you can use a binomial distribution, check the BINS acronym. First, the outcomes must be Binary (only two possibilities, like pass/fail or heads/tails). Second, the trials must be Independent, meaning one result does not influence another. Third, there must be a fixed Number of trials, which we call $n$ . Finally, the probability of Success, called $p$ , must remain the exact same for every single trial.

The Formula

If a scenario passes the BINS test, you can calculate the probability of getting exactly $k$ successes out of $n$ trials using this formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ . In this formula, $\binom{n}{k}$ calculates the number of different ways your successes can be arranged among the trials. The $p^k$ part is the probability of the successes happening, and $(1-p)^{n-k}$ is the probability of the remaining trials being failures.

Where students slip up

The most common mistake happens when translating word problems into math. Students often calculate the probability for exactly $k$ successes when the question asks for "at least" or "at most" $k$ successes. If a question asks for the probability of "at least 3 successes" in 5 trials, you cannot just calculate $P(X = 3)$ . You must calculate the probabilities for 3, 4, and 5, and add them all together.

Worked through

A basketball player makes $80\%$ of her free throws. If she shoots 5 free throws in a game, what is the probability she makes exactly 4 of them?

First, we identify our variables. The number of trials is $n = 5$ . The probability of success on any one shot is $p = 0.80$ . The number of successes we want to find the probability for is $k = 4$ . The probability of failure is $1 - 0.80 = 0.20$ .

Now we plug these into the binomial formula: $P(X = 4) = \binom{5}{4} (0.80)^4 (0.20)^{5-4}$

First, calculate the combinations: $\binom{5}{4} = 5$ . Then calculate the probabilities: $(0.80)^4 = 0.4096$ and $(0.20)^1 = 0.20$ .

Multiply them all together: $5 \times 0.4096 \times 0.20 = 0.4096$ .

There is a $40.96\%$ chance she will make exactly 4 out of 5 free throws.

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Where this comes from: OpenStax Introductory Statistics, Chapter 4: Discrete Random Variables · Khan Academy, Probability Unit: Binomial probability