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Some randomly selected high school students were asked to name their favorite sport to watch. The table displays the distribution of results.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Sport & Football & Basketball & Baseball & Soccer & None \\
\hline
Probability & 0.23 & 0.18 & 0.26 & 0.17 & 0.16 \\
\hline
\end{tabular}

What is the probability that a student chose football given that they like watching sports?

A. 0.16
B. 0.23
C. 0.27
D. 0.77


Sagot :

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
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