To determine the conditional probability [tex]\(P(A \mid B)\)[/tex], which is the probability that a randomly selected student is in the chess club given that they are in the karate club, we can use the formula for conditional probability:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here, [tex]\(P(A \cap B)\)[/tex] is the probability that a student is in both the chess club and the karate club, and [tex]\(P(B)\)[/tex] is the probability that a student is in the karate club.
Based on the information provided:
- There are 10 students in total.
- 4 students are in the chess club.
- 3 students are in the karate club.
- 2 students are in both clubs.
First, we need to calculate [tex]\(P(A \cap B)\)[/tex]:
[tex]\[ P(A \cap B) = \frac{\text{Number of students in both clubs}}{\text{Total number of students}} = \frac{2}{10} = 0.2 \][/tex]
Next, we calculate [tex]\(P(B)\)[/tex]:
[tex]\[ P(B) = \frac{\text{Number of students in the karate club}}{\text{Total number of students}} = \frac{3}{10} = 0.3 \][/tex]
Now, we can calculate [tex]\(P(A \mid B)\)[/tex]:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.3} \][/tex]
[tex]\[ P(A \mid B) = \frac{2}{3} \][/tex]
Converting [tex]\(\frac{2}{3}\)[/tex] to a decimal:
[tex]\[ \frac{2}{3} \approx 0.6667 \][/tex]
Thus, the conditional probability [tex]\(P(A \mid B)\)[/tex] is approximately [tex]\(0.6667\)[/tex]. The given answer of [tex]\(\frac{2}{10} = 0.20\)[/tex] is incorrect.
