To determine the probability that a student chose football given that they like watching sports, we follow the steps needed to calculate the conditional probability. Here’s the step-by-step solution:
1. Identify the given probabilities:
- Probability of choosing Football, [tex]\( P(\text{Football}) = 0.23 \)[/tex]
- Probability of choosing Basketball, [tex]\( P(\text{Basketball}) = 0.18 \)[/tex]
- Probability of choosing Baseball, [tex]\( P(\text{Baseball}) = 0.26 \)[/tex]
- Probability of choosing Soccer, [tex]\( P(\text{Soccer}) = 0.17 \)[/tex]
- Probability of not choosing a sport (None), [tex]\( P(\text{None}) = 0.16 \)[/tex]
2. Calculate the total probability of choosing any sport:
[tex]\[
P(\text{Any sport}) = P(\text{Football}) + P(\text{Basketball}) + P(\text{Baseball}) + P(\text{Soccer})
= 0.23 + 0.18 + 0.26 + 0.17
= 0.84
\][/tex]
3. Use the formula for conditional probability:
The conditional probability that a student chose football given that they like watching sports is given by:
[tex]\[
P(\text{Football} \mid \text{Sport}) = \frac{P(\text{Football})}{P(\text{Any sport})}
\][/tex]
4. Substitute the known values into the formula:
[tex]\[
P(\text{Football} \mid \text{Sport}) = \frac{0.23}{0.84}
\approx 0.2738
\][/tex]
5. Round the result to two decimal places (if needed):
[tex]\[
P(\text{Football} \mid \text{Sport}) = 0.27
\][/tex]
Thus, the probability that a student chose football given that they like watching sports is approximately 0.27, which corresponds to the given answer choices.
Hence, the correct answer is:
0.27
