Back to Apply the general Multiplication Rule — Problem 1 · Task Set 25

Exercises: Apply the General Multiplication Rule

Work through each section in order. Use the general Multiplication Rule, $P(A \text{ and } B) = P(A) \cdot P(B \mid A)$, for joint probabilities. For draws WITHOUT replacement, remember that BOTH the favorable count and the total shrink after the first draw. On a tree diagram, MULTIPLY along a path and ADD across paths. Leave fractions exact unless told otherwise.

Grade 11·21 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-cp-b-8

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Warm-Up: Product Rule and Conditional Probability

These problems review the prior ideas the new rule builds on.

For independent events, the product rule from earlier is $P(A \text{ and } B) = P(A) \cdot P(B)$ . Two cards are drawn from a 52-card deck without replacement. Why might this simple product rule give the wrong joint probability here?

Because the draws are dependent: removing the first card changes the probabilities for the second draw, so the second factor is no longer the original $P(B)$ .

Because probabilities can never be multiplied when cards are involved.

Because the deck has too many cards for the rule to apply.

Because $P(A) \cdot P(B)$ always overcounts by exactly one card.