A survey is conducted to study the favorite dessert of individuals in different regions. The two-way table is given below:
| | Cake | Cookies | Ice Cream | Total |
| :------ | :--- | :------ | :-------- | :---- |
| South | 8 | 9 | 16 | 33 |
| Midwest | 20 | 4 | 10 | 34 |
| North | 18 | 11 | 4 | 33 |
| Total | 46 | 24 | 30 | 100 |
What is the probability that a randomly selected person from this survey is from the South, given their favorite dessert is ice cream?
[tex]P( \text{South} \mid \text{Ice Cream}) = [?] \%[/tex]
Round your answer to the nearest whole percent.
Answer (2)
Define events: South (S) and Ice Cream (IC).
Use conditional probability formula: P ( S ∣ I C ) = P ( I C ) P ( S ∩ I C ) .
Calculate probabilities: P ( S ∩ I C ) = 100 16 and P ( I C ) = 100 30 .
Find the conditional probability and convert to percentage: P ( S ∣ I C ) = 30 16 ≈ 53% .
Explanation
Understand the problem and provided data We are given a two-way table that summarizes the favorite dessert of individuals in different regions. We want to find the probability that a randomly selected person is from the South, given that their favorite dessert is ice cream. This is a conditional probability problem.
Define events Let S be the event that a person is from the South. Let IC be the event that a person's favorite dessert is ice cream. We want to find P ( S ∣ I C ) , which is the probability of a person being from the South given that they like ice cream.
State the formula for conditional probability The formula for conditional probability is: P ( S ∣ I C ) = P ( I C ) P ( S ∩ I C )
Calculate the probabilities From the table, the number of people from the South who like ice cream is 16. The total number of people surveyed is 100. Therefore, P ( S ∩ I C ) = 100 16 From the table, the total number of people who like ice cream is 30. The total number of people surveyed is 100. Therefore, P ( I C ) = 100 30
Calculate the conditional probability and convert to percentage Now we can calculate the conditional probability: P ( S ∣ I C ) = P ( I C ) P ( S ∩ I C ) = 100 30 100 16 = 30 16 To express this as a percentage, we multiply by 100: 30 16 × 100 = 53.333... Rounding to the nearest whole percent, we get 53%.
State the final answer Therefore, the probability that a randomly selected person from this survey is from the South, given their favorite dessert is ice cream is approximately 53%.
Examples
Conditional probability is used in many real-world scenarios. For example, in medical diagnosis, we might want to know the probability that a patient has a certain disease given that they have a specific symptom. In marketing, we might want to know the probability that a customer will buy a product given that they have visited our website. In finance, we might want to know the probability that a loan will default given certain economic conditions. These probabilities help in making informed decisions.
The probability that a randomly selected person from this survey is from the South, given their favorite dessert is ice cream, is approximately 53%. This is computed using the conditional probability formula. Gaining this insight helps understand how preferences vary by region.
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