Question

identify the error in finding the probability of randomly drawing the given card from a standard deck of 52 cards.

$p(\text{club or } 9)$
$= p(\text{club}) + p(9) + p(\text{club and } 9)$
$= \frac{13}{52} + \frac{4}{52} + \frac{1}{52} = \frac{9}{26}$

Explanation:

Step1: Recall Probability Formula

For mutually inclusive events, \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \).

Step2: Identify Error

The given calculation used \( + P(\text{club and } 9) \) instead of \( - P(\text{club and } 9) \). The correct formula subtracts the probability of the intersection (to avoid double - counting the 9 of clubs).

Step3: Correct Calculation (Optional for Error Identification)

\( P(\text{club})=\frac{13}{52} \), \( P(9)=\frac{4}{52} \), \( P(\text{club and } 9)=\frac{1}{52} \).
Using the correct formula: \( P(\text{club or } 9)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4-1}{52}=\frac{16}{52}=\frac{4}{13} \), while the wrong calculation added instead of subtracting the intersection probability.

Answer:

The error is in the formula for \( P(\text{club or } 9) \); it should be \( P(\text{club})+P(9)-P(\text{club and } 9) \) (subtract \( P(\text{club and } 9) \), not add it) to avoid double - counting the 9 of clubs.