Independence

Learn independent events, dependent draws.

Lesson 3 of 32 parts Practice Flashcards

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Independent Events

Objective: Explain independent events, apply the stated method, and check its conditions.

Events are independent when P(A∩B)=P(A)P(B), equivalently conditioning on one does not change the other’s probability when the condition has positive probability. Independence is not the same as mutual exclusivity; disjoint positive-probability events cannot be independent. Mutual exclusivity and independence are different relationships.

Formulas

P(A∩B)=P(A)P(B) for independent events

Worked examples

Try itIndependent events have P(A)=1/6 and P(B)=1/7. Find P(A∩B).Show answer

Step 1: For independent events, multiply probabilities. Step 2: Use (1/6)(1/7). Step 3: P(A∩B)=1/42.

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Independent Events

Objective: Explain independent events, apply the stated method, and check its conditions.

Formulas

P(A∩B)=P(A)P(B) for independent events

Worked examples

Try itIndependent events have P(A)=1/6 and P(B)=1/7. Find P(A∩B).Show answer

Step 1: For independent events, multiply probabilities. Step 2: Use (1/6)(1/7). Step 3: P(A∩B)=1/42.

Dependent Draws

Objective: Explain dependent draws, apply the stated method, and check its conditions.

Sampling without replacement changes the remaining composition after each draw. Multiply the first-event probability by the second probability conditioned on the first. The denominators shrink and relevant favorable counts also change; independent-draw multiplication is generally inappropriate. Without-replacement draws generally are dependent.

Formulas

$P(A then B)=P(A)P(B|A)$

Worked examples

Try itA bag holds 6 red and 7 blue counters. Two are drawn without replacement. Find P(both red).Show answer

Step 1: First-red probability is 6/13. Step 2: After a red, the second-red probability is 5/12. Step 3: Multiply to get 5/26; the draws are dependent.

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