A survey is conducted to study the favorite sport of individuals in different age groups. The two-way table is given below:
| | Football | Basketball | Baseball | Total |
| :------------------ | :------- | :--------- | :------- | :---- |
| $8 - 12 ~ yrs$ | 10 | 12 | 10 | $32$ |
| $13 - 17 ~ yrs$ | 8 | 6 | 24 | $38$ |
| $18 - 22 ~ yrs$ | 16 | 2 | 12 | $30$ |
| Total | $34$ | $20$ | $46$ | $100$ |
What is the probability that a randomly selected person from this survey is 13 to 17 years old, given their favorite sport is football?
$P(13-17 yrs | \text { Football })=[?] %$
Round your answer to the nearest whole percent.
Answer (2)
Define events A (13-17 years old) and B (favorite sport is football).
Calculate P ( A ∩ B ) = 100 8 and P ( B ) = 100 34 from the table.
Use the conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) = 34 8 .
Convert to percentage and round: 34 8 × 100 ≈ 24 % .
Explanation
Understand the problem We are given a two-way table that shows the favorite sport of individuals in different age groups. The question asks for the probability that a randomly selected person is 13 to 17 years old, given that their favorite sport is football. This is a conditional probability problem.
Define events and the conditional probability formula Let A be the event that a randomly selected person is 13-17 years old, and let B be the event that a randomly selected person's favorite sport is football. We want to find P ( A ∣ B ) , the probability of A given B. The formula for conditional probability is: P ( A ∣ B ) = P ( B ) P ( A ∩ B )
Find the probabilities from the table From the table, we can see that the number of people who are 13-17 years old and prefer football is 8. The total number of people surveyed is 100. Therefore, the probability of a person being 13-17 years old and preferring football is: P ( A ∩ B ) = 100 8 Also, from the table, the total number of people who prefer football is 34. Therefore, the probability of a person preferring football is: P ( B ) = 100 34
Calculate the conditional probability Now we can calculate the conditional probability: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) = 100 34 100 8 = 34 8 To express this as a percentage, we multiply by 100: 34 8 × 100 = 34 800 ≈ 23.529
Round to the nearest whole percent Rounding the percentage to the nearest whole percent, we get 24%.
Examples
Conditional probability is used in many real-world scenarios, such as medical diagnosis, risk assessment, and marketing. For example, a doctor might want to know the probability that a patient has a certain disease, given that they have certain symptoms. Or a marketer might want to know the probability that a customer will buy a product, given that they have visited a certain website. In this case, we found the probability that a randomly selected person is 13 to 17 years old, given that their favorite sport is football. This type of analysis can help understand the preferences of different age groups.
The probability that a randomly selected person is 13 to 17 years old given their favorite sport is football is approximately 24%. This is calculated using the conditional probability formula and the values obtained from a two-way table. Rounding the result gives the final answer.
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