Applying the Multiplication Rule for Probability - Statistics and Probability

1. In a uniform model with $\#(S)=50$, $\#(A)=20$, $\#(A \cap B)=5$, find $P(A \cap B)$.

   $\frac{1}{10}$. Count favorable outcomes over total: $\frac{5}{50} = \frac{1}{10}$.

2. Which expression equals the probability that both events occur: $P(A \text{ and } B)$?

   $P(A \cap B)$. "And" in probability means intersection.

3. What is $P(A \cap B)$ if $P(B)=0.20$ and $P(A\mid B)=0.50$?

   $0.10$. Use $P(A \cap B) = P(B)P(A|B) = 0.20 \times 0.50$.

4. State the general Multiplication Rule for the probability of $A \cap B$.

   $P(A \cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$. Expresses joint probability using conditional probability.

5. If $P(A \cap B)=0$, what must be true about $A$ and $B$?

   $A$ and $B$ are mutually exclusive (cannot occur together). Zero intersection means events have no common outcomes.

6. What condition makes the simplified rule $P(A \cap B)=P(A)P(B)$ valid?

   $A$ and $B$ are independent. Independence allows factoring joint probability into product.

7. In a uniform probability model, how do you compute $P(B\mid A)$ from counts?

   $P(B\mid A)=\frac{\#(A \cap B)}{\#(A)}$. Restricts sample space to outcomes where $A$ occurred.

8. State the definition of conditional probability $P(B\mid A)$ in terms of $P(A \cap B)$ and $P(A)$.

   $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$. Ratio of joint probability to marginal probability of $A$.

9. Find $P(A \cap B)$ given $P(A)=0.4$ and $P(B\mid A)=0.25$.

   $P(A \cap B)=0.10$. Apply multiplication rule: $0.4 \times 0.25 = 0.10$.

10. In a uniform probability model, how do you compute $P(E)$ from counts?

    $P(E)=\frac{\#(E)}{\#(S)}$. Uniform model means all outcomes are equally likely.

11. In a uniform model, what is $P(B\mid A)$ in terms of counts $|A\cap B|$ and $|A|$?

    $P(B\mid A)=\frac{|A\cap B|}{|A|}$. Count outcomes in both $A$ and $B$ divided by outcomes in $A$.

12. Two fair coins are flipped. Let $A=\{\text{first is H}\}$ and $B=\{\text{exactly one H}\}$. Find $P(A \cap B)$.

    $P(A \cap B)=\frac{1}{4}$. Only HT satisfies both conditions out of 4 outcomes.

13. Two fair coins are flipped. Let $A=\{\text{first is H}\}$ and $B=\{\text{exactly one H}\}$. Find $P(B\mid A)$.

    $P(B\mid A)=\frac{1}{2}$. Given first H, only HT has exactly one H out of HH, HT.

14. In a uniform model, if $|A|=10$ and $|A\cap B|=4$, find $P(B\mid A)$.

    $P(B\mid A)=\frac{4}{10}=0.4$. Four outcomes satisfy both conditions out of 10 in $A$.

15. Compute $P(B\mid A)$ given $P(A \cap B)=0.12$ and $P(A)=0.3$.

    $P(B\mid A)=0.4$. Divide joint probability by marginal: $0.12 \div 0.3 = 0.4$.

16. Which formula gives $P(B\mid A)$ in terms of $P(A \cap B)$ and $P(A)$?

    $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$. Rearranges the multiplication rule to isolate conditional probability.

17. In a uniform probability model with $N$ equally likely outcomes, what is $P(E)$ for an event with $|E|$ outcomes?

    $P(E)=\frac{|E|}{N}$. Favorable outcomes divided by total outcomes in uniform model.

18. Identify the independence criterion stated using conditional probability.

    $P(B\mid A)=P(B)$ (with $P(A)>0$). Independence means conditioning doesn't change probability.

19. What equivalent Multiplication Rule expression gives $P(A \cap B)$ starting with $P(B)$?

    $P(A \cap B)=P(B)P(A\mid B)$. Alternative form using $P(B)$ first, then $P(A|B)$.

20. Which formula gives $P(A \cap B)$ using $P(A)$ and $P(B\mid A)$?

    $P(A \cap B)=P(A)P(B\mid A)$. Direct application of the multiplication rule.

21. Find $P(A \cap B)$ given $P(B)=0.3$ and $P(A\mid B)=0.5$.

    $P(A \cap B)=0.15$. Apply multiplication rule: $0.3 \times 0.5 = 0.15$.

22. In a uniform model with $N=20$, if $|A|=8$ and $|A\cap B|=2$, find $P(A \cap B)$.

    $P(A \cap B)=\frac{2}{20}=0.10$. Two favorable outcomes out of 20 total outcomes.

23. In a uniform model with $\#(A)=20$ and $\#(A \cap B)=5$, find $P(B\mid A)$.

    $\frac{1}{4}$. Count $B$ outcomes within $A$: $\frac{5}{20} = \frac{1}{4}$.

24. A fair die is rolled. Let $A=\{\text{even}\}$ and $B=\{\ge 5\}$. Find $P(B\mid A)$.

    $P(B\mid A)=\frac{1}{3}$. Given even (2,4,6), only 6 satisfies $\ge 5$.

25. Find and correct the error: "$P(A \cap B)=P(A)+P(B\mid A)$".

    Correct: $P(A \cap B)=P(A)P(B\mid A)$. Should multiply, not add, for joint probability.

26. If $A$ and $B$ are independent, what does the Multiplication Rule simplify to?

    $P(A \cap B)=P(A)P(B)$. Independence means conditional equals marginal probability.

27. What does the notation $P(B\mid A)$ mean in words?

    Probability that $B$ occurs given that $A$ occurred. The vertical bar means "given" or "conditional on."

28. Identify the required condition for $P(B\mid A)=\frac{P(A \cap B)}{P(A)}$ to be defined.

    $P(A)>0$. Division by zero is undefined when $P(A)=0$.

29. Identify the correct interpretation of $P(A)P(B\mid A)$ in context.

    Probability $A$ occurs, then $B$ occurs given $A$ occurred. Multiplication rule gives sequential probability interpretation.

30. What is $P(A \cap B)$ if $P(A)=\frac{1}{4}$ and $P(B\mid A)=\frac{1}{2}$?

    $\frac{1}{8}$. Apply multiplication rule: $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$.

31. Compute $P(A\mid B)$ given $P(A \cap B)=0.18$ and $P(B)=0.6$.

    $P(A\mid B)=0.3$. Divide joint probability by marginal: $0.18 \div 0.6 = 0.3$.

32. In a uniform model with $\#(S)=30$, $\#(A)=12$, $\#(B)=10$, $\#(A \cap B)=4$, find $P(A)P(B\mid A)$.

    $\frac{2}{5}\cdot\frac{1}{3}=\frac{2}{15}$. $P(A)=\frac{12}{30}=\frac{2}{5}$, $P(B|A)=\frac{4}{12}=\frac{1}{3}$.

33. A deck has 52 cards. Draw 1 card. Let $A=\{\text{heart}\}$ and $B=\{\text{face card}\}$. Find $P(A \cap B)$.

    $P(A \cap B)=\frac{3}{52}$. Three face cards (J,Q,K) in hearts out of 52 cards.

34. What is $P(B\mid A)$ if $P(A \cap B)=0.12$ and $P(A)=0.30$?

    $0.40$. Divide joint probability by marginal: $0.12 \div 0.30 = 0.40$.

35. A fair die is rolled. Let $A=\{\text{even}\}$ and $B=\{\ge 5\}$. Find $P(A \cap B)$.

    $P(A \cap B)=\frac{1}{6}$. Only outcome 6 is both even and $\ge 5$.

36. State the general Multiplication Rule for the intersection $P(A \cap B)$ using conditional probability.

    $P(A \cap B)=P(A)P(B\mid A)$. Multiply $P(A)$ by the probability of $B$ given $A$ occurred.

37. A deck has 52 cards. Draw 1 card. Let $A=\{\text{heart}\}$ and $B=\{\text{face card}\}$. Find $P(B\mid A)$.

    $P(B\mid A)=\frac{3}{13}$. Three face cards out of 13 hearts in the suit.

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