Conditional Probability — Practice Quiz

A Probability cheat sheet for Conditional Probability — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Conditional Probability

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Where: \(P(B) \neq 0\)

Bayes' Theorem

$$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$$

Law of Total Probability

$$P(A) = \sum_{i} P(A|B_i) \cdot P(B_i)$$

Where: \(B_i\) partition the sample space

Practice quiz

  1. If $P(A \cap B) = 0.2$ and $P(B) = 0.5$, what is $P(A|B)$?

    • $0.1$
    • $0.4$
    • $0.25$
    • $0.7$

    Answer: $0.4$

  2. Given $P(B_1) = 0.3$, $P(B_2) = 0.7$, $P(A|B_1) = 0.6$, and $P(A|B_2) = 0.2$, where $B_1$ and $B_2$ form a partition of the sample space. What is $P(A)$?

    • $0.8$
    • $0.24$
    • $0.32$
    • $0.42$

    Answer: $0.32$

  3. If $P(A|B) = 0.7$, $P(B) = 0.4$, and $P(A) = 0.5$, what is $P(B|A)$?

    • $0.28$
    • $0.56$
    • $0.35$
    • $0.8$

    Answer: $0.56$

  4. For two events $A$ and $B$, if $P(A|B) = P(A)$, what can be concluded about $A$ and $B$?

    • $A$ and $B$ are mutually exclusive
    • $A$ and $B$ are independent
    • $A$ is a subset of $B$
    • $B$ is a subset of $A$

    Answer: $A$ and $B$ are independent

  5. The Law of Total Probability requires that the events $B_i$ form a partition of the sample space. Which of the following conditions is NOT necessarily true for a partition?

    • The events $B_i$ are mutually exclusive
    • The union of all $B_i$ covers the entire sample space
    • $P(B_i) > 0$ for all $i$
    • The events $B_i$ are independent

    Answer: The events $B_i$ are independent

  6. A factory produces items using two machines, $M_1$ and $M_2$. $M_1$ produces $60\%$ of the items and $M_2$ produces $40\%$. $2\%$ of items from $M_1$ are defective, and $3\%$ of items from $M_2$ are defective. If an item is found to be defective, what is the probability it came from $M_1$?

    • $0.6$
    • $0.5$
    • $0.024$
    • $0.4$

    Answer: $0.5$

  7. Which of the following expressions is equivalent to $P(A \cap B)$?

    • $P(A|B)P(A)$
    • $P(B|A)P(B)$
    • $P(A|B)P(B)$
    • $P(A) + P(B)$

    Answer: $P(A|B)P(B)$

  8. Bayes' Theorem is primarily used to calculate:

    • The probability of the intersection of two events
    • The probability of the union of two events
    • The posterior probability of an event given new evidence
    • The marginal probability of an event

    Answer: The posterior probability of an event given new evidence

  9. Suppose there are three disjoint and exhaustive events $B_1, B_2, B_3$ such that $P(B_1) = 0.2$, $P(B_2) = 0.5$, $P(B_3) = 0.3$. If $P(A|B_1) = 0.1$, $P(A|B_2) = 0.4$, and $P(A|B_3) = 0.3$, what is $P(A)$?

    • $0.8$
    • $0.31$
    • $0.25$
    • $0.4$

    Answer: $0.31$

  10. In a certain population, $10\%$ of people have a specific genetic marker ($M$). A test for this marker is $95\%$ accurate for those who have the marker (true positive rate) and $90\%$ accurate for those who do not have the marker (true negative rate). If a person tests positive for the marker, what is the probability they actually have the marker?

    • $0.95$
    • $0.185$
    • $0.5135$
    • $0.095$

    Answer: $0.5135$

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