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High school · Probability

Count successes with a binomial model

Explore the exact probability of each success count instead of relying on a random sample.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Count successes with a binomial model. Selected success count k: 0. P(X = k): 0.39%. P(X ≤ k): 0.39%. Mean count np: 4Exact probabilities · scanning k = 00.27008Horizontal: successes · Vertical: probability
Trials are independent, have two outcomes, and share a constant success probability. The number of trials is fixed beforehand. The vertical scale adjusts to the largest probability so small bars remain visible; bar height is not a count of people or experiments.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Selected success count k
0
P(X = k)
0.39%
P(X ≤ k)
0.39%
Mean count np
4

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

For a fixed number of independent trials with the same success probability, many different orders can lead to the same total. The binomial coefficient counts those orders, while the powers of p and 1−p give each order's probability.

A relationship to keep

P(X = k) = C(n, k) pᵏ(1−p)ⁿ⁻ᵏ

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Specify the experiment

    Choose the fixed number of trials and the probability of success on each trial. Success is simply the outcome you are counting; it need not mean winning.

  2. STEP 2

    Inspect each count

    Playback scans the possible counts k = 0 through n. Each bar is an exact model probability, not a simulated frequency. The amber bar marks the count currently being inspected.

  3. STEP 3

    Accumulate probability

    The teal bars up to k add to P(X ≤ k). At the final count, their probabilities total one. The mean np may fall between integer counts and need not be an outcome you can observe.

Your turn to explain

Make a prediction. Test your reasoning.

For n = 4 and p = 0.5, what is the chance of exactly two successes?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

6/16 = 0.375, or 37.5%. There are C(4, 2) = 6 orders with two successes, each with probability (0.5)⁴ = 1/16.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Intermediate · Statistics

How often will a small batch include a false alarm?

Check the binomial conditions, use a complement for at least one event, and calculate an exact count probability.

Developing · Statistics

When can you count outcomes to find a probability?

Build a sample space, count overlapping events, and avoid assuming outcomes are equally likely without justification.