Below a two-way table is given for student activities:
| | Sports | Drama | Work | Total |
| -------- | ------ | ----- | ---- | ----- |
| Sophomore | 20 | 7 | 3 | 30 |
| Junior | 20 | 13 | 2 | 35 |
| Senior | 25 | 5 | 5 | 35 |
| Total | 65 | 25 | 10 | 100 |
Follow the steps to find the probability a student is in sports, given that they are a senior.
Use the formula below to get your final answer.
[tex]$\begin{array}{c}
P(\text { senior })=0.35 \quad P(\text { senior and sports })=0.25 \
P(\text { sports } \mid \text { senior })=\frac{P(\text { sports and senior })}{P(\text { senior })}=[?] %\
\end{array}$[/tex]
Round your answer to the nearest whole percent.
Answer (2)
Use the conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A and B ) .
Substitute the given probabilities: P ( sports ∣ senior ) = 0.35 0.25 .
Calculate the result: 0.35 0.25 ≈ 0.7143 .
Convert to percentage and round: 0.7143 × 100 ≈ 71% .
Explanation
Understand the problem and provided data We are given the probabilities P ( senior ) = 0.35 and P ( senior and sports ) = 0.25 . We want to find the probability that a student is in sports, given that they are a senior. This is a conditional probability, which we can denote as P ( sports ∣ senior ) .
State the conditional probability formula We use the formula for conditional probability: P ( sports ∣ senior ) = P ( senior ) P ( sports and senior ) .
Substitute the given values Substitute the given values into the formula: P ( sports ∣ senior ) = 0.35 0.25 .
Calculate the result Perform the division: 0.35 0.25 = 0.7142857142857143 .
Convert to percentage To express this as a percentage, we multiply by 100: 0.7142857142857143 × 100 = 71.42857142857143% .
Round to the nearest whole percent Finally, we round the percentage to the nearest whole percent: 71.42857142857143% ≈ 71% .
State the final answer Therefore, the probability that a student is in sports, given that they are a senior, is approximately 71% .
Examples
Conditional probability is used in many real-world scenarios. For example, in medical diagnosis, it helps determine the probability of a disease given certain symptoms. In marketing, it can predict the likelihood of a customer making a purchase based on their browsing history. In finance, it's used to assess the risk of investments based on various market conditions. Understanding conditional probability allows us to make more informed decisions in these and many other areas.
The probability that a senior student is involved in sports is calculated using conditional probability. Using the values provided, we find that approximately 71% of seniors participate in sports. This is derived from the formula P ( sports ∣ senior ) = P ( senior ) P ( sports and senior ) and rounding the result to the nearest whole percent.
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