Back to Exercise: Find conditional probability from a model

Exercises: Find Conditional Probability as the Fraction of B's Outcomes That Are Also A

Work through each section in order. To find P(A | B), restrict to B's
outcomes first, then count how many of them are also in A: the denominator
is the number of B's outcomes, NOT all outcomes. Write probabilities as
fractions in lowest terms unless asked otherwise, and interpret each
conditional probability with a sentence: "among the B's, this fraction are A."

Grade 10·20 problems·~32 min·Common Core Math - HS Statistics and Probability·group·hss-cp-b-6

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Warm-Up: Sample Spaces and the Conditional Formula

These problems review ideas you already know from Cluster A.

A standard die is rolled, $S = \{1, 2, 3, 4, 5, 6\}$ . You are asked for $P(\text{even} \mid \text{greater than } 3)$ . When you condition on "greater than 3," what becomes the new sample space?

$\{4, 5, 6\}$ — only the outcomes greater than 3

$\{1, 2, 3, 4, 5, 6\}$ — all six outcomes, unchanged

$\{2, 4, 6\}$ — the even outcomes

$\{4, 6\}$ — the even outcomes greater than 3

Recall the conditional-probability formula from HSS.CP.A.3. For events $A$ and $B$ with $P(B) > 0$ , which expression equals $P(A \mid B)$ ?

$\dfrac{P(A \text{ and } B)}{P(B)}$

$\dfrac{P(A \text{ and } B)}{P(A)}$

$P(A) \cdot P(B)$

$\dfrac{P(B)}{P(A \text{ and } B)}$

A standard die is rolled, $S = \{1, 2, 3, 4, 5, 6\}$ . Let $B = $ "greater than 3" $= \{4, 5, 6\}$ . By restricting to $B$ , how many outcomes are in the new sample space? ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ Of those, how many are even? ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

Fluency Practice

For each, restrict to B and count A among B's outcomes. Enter probabilities as fractions in lowest terms (for example, 1/3).

A fair die is rolled. Let $A = $ "even" and $B = $ "greater than 3" $= \{4, 5, 6\}$ . Find $P(A \mid B)$ , the probability of an even number given the roll is greater than 3. Enter your answer as a fraction.

A fair die is rolled. Let $A = $ "even" $= \{2, 4, 6\}$ and $B = $ "greater than 2" $= \{3, 4, 5, 6\}$ . Find $P(A \mid B)$ . Enter your answer as a fraction in lowest terms.

One card is drawn from a standard 52-card deck. Let $A = $ "a king" and $B = $ "a face card" (Jack, Queen, or King). Find $P(A \mid B)$ , the probability the card is a king given it is a face card. Enter your answer as a fraction in lowest terms.

One card is drawn from a standard 52-card deck. Verify the counting result with the HSS.CP.A.3 formula. Using $P(\text{king and face}) = \frac{4}{52}$ and $P(\text{face}) = \frac{12}{52}$ , compute $\dfrac{P(\text{king and face})}{P(\text{face})}$ . Enter your answer as a fraction in lowest terms.

A survey of 40 people is summarized in the two-way table. Conditioning on a row or column restricts to that group.

	Coffee	Tea	Total
Adult	18	6	24
Teen	4	12	16
Total	22	18	40

Find $P(\text{Coffee} \mid \text{Adult})$ — among adults, the fraction who prefer coffee. Enter your answer as a fraction in lowest terms.

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