Answers: The General Multiplication Rule

    • (a) Headaches: 1/6 = 16.67%; Took the drug: 25%; Both: 50/1200 = 4.167%.

    • (b) Yes, because $P(\text{headaches and drug taken}) = 4.167\%$, and $P(\text{headaches}) \cdot P(\text{drug taken}) = 4.167\%$.

    • (c) Some of the assumptions include:

      • We assume that this survey is representative of all people who might take this drug;

      • We assume there are no other factors affecting whether someone takes this drug and whether they experience headaches (e.g., maybe there’s another drug they also tend to take).

  2. $P(B|A) = \frac{0.4}{0.6} = \frac{2}{3}$. More information would be needed to determine $P(B)$, e.g., knowing $P(B | A^c)$ would be enough information.

  3. $P(\text{Second card is an ace})=\frac{4}{52}$

    • $P(A)= \frac{13}{52} \cdot \frac{12}{51}$

    • $P(B) = \frac{12}{51}$

    • $P(C) = \frac{3}{51}$

    • Events $A$ and $B$ are not independent, as $P(AB) = P(A)$, which is not equal to $P(A)\cdot P(B)$. Events $B$ and $C$ are not independent, either, as $P(BC)=0$ — the events are mutually exclusive!

  5. $P(\text{Results are different}) = \frac{5}{6}$

  6. $P(\text{Results are different}) = \frac{5}{6} \cdot \frac{4}{6}$