Multiply Independent Event Probabilities

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A fair coin and a fair 3-section spinner are independent. What is the chance of heads and section 1? Candidate answers: (A) 1/2 (B) 1/3 (C) 1/6 (D) 5/6.
• Independent check inputs: A fair coin and a fair 4-section spinner are independent. What is the chance of heads and section 1? Candidate answers: (A) 1/2 (B) 1/4 (C) 1/8 (D) 3/4.

Learn the idea

A complete population, a selected sample and a probability model are different sources of information. Sampling supports estimates, observed trial counts support experimental proportions, and explicit independence permits multiplication of model probabilities. Independent events have a joint probability equal to the product of their individual probabilities. A coin-and-spinner outcome can also be checked by enumerating equally likely ordered pairs.

Here is the method on the supplied teaching case. Heads has chance 1/2 and the named section has chance 1/3. Independence permits multiplication: 1/2×1/3=1/6.

Independence is a stated assumption here, not something true of every pair of events. An and question uses the joint event; adding probabilities would count different outcomes and can exceed one.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: A fair coin and a fair 10-section spinner are independent. What is the chance of heads and section 1? Candidate answers: (A) 1/10 (B) 1/2 (C) 1/20 (D) 3/5. Show the mathematical reason, not only the selected answer. Expected result: 1/20. Heads has chance 1/2 and the named section has chance 1/10. Independence permits multiplication: 1/2×1/10=1/20.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.