A taste test asks people from Texas and California which pasta they prefer, brand A or brand B. This table shows the results.
| | Brand A | Brand B | Total |
| :---------- | :------ | :------ | :---- |
| Texas | 80 | 45 | 125 |
| California | 90 | 60 | 150 |
| Total | 170 | 105 | 275 |
A person is randomly selected from those tested. What is the probability that the person is from Texas, given that the person prefers brand A? Round your answer to two decimal places.
A. 0.45
B. 0.64
C. 0.47
D. 0.62
Answer (2)
Define events: T (person from Texas), A (person prefers brand A).
Use conditional probability formula: P ( T ∣ A ) = P ( A ) P ( T ∩ A ) .
Calculate P ( T ∩ A ) = 275 80 and P ( A ) = 275 170 .
Find P ( T ∣ A ) = 170 80 ≈ 0.47 .
Explanation
Understand the problem and provided data We are given a table that shows the results of a taste test between brand A and brand B pasta preferences for people from Texas and California. We want to find the probability that a randomly selected person is from Texas, given that they prefer brand A.
Define events Let T be the event that the person is from Texas. Let A be the event that the person prefers brand A. We want to find P(T|A), which is the probability that the person is from Texas given that they prefer brand A.
State the conditional probability formula We will use the formula for conditional probability: P ( T ∣ A ) = P ( A ) P ( T ∩ A ) .
Identify the necessary values from the table From the table, the number of people from Texas who prefer brand A is 80. The total number of people who prefer brand A is 170. The total number of people tested is 275.
Calculate probabilities We can calculate P ( T ∩ A ) as the number of people from Texas who prefer brand A divided by the total number of people tested: P ( T ∩ A ) = 275 80 .
We can calculate P ( A ) as the total number of people who prefer brand A divided by the total number of people tested: P ( A ) = 275 170 .
Apply the conditional probability formula Now we can calculate the conditional probability: P ( T ∣ A ) = P ( A ) P ( T ∩ A ) = 275 170 275 80 = 170 80 = 17 8 .
Calculate and round the final answer To round the answer to two decimal places, we divide 8 by 17: 17 8 ≈ 0.470588... Rounding to two decimal places, we get 0.47.
State the final answer Therefore, the probability that the person is from Texas, given that the person prefers brand A is approximately 0.47.
Examples
Conditional probability is used in many real-world scenarios, such as medical diagnosis, risk assessment, and marketing. For example, in medical diagnosis, doctors use conditional probability to determine the probability of a patient having a disease given that they have certain symptoms. In marketing, companies use conditional probability to determine the probability of a customer buying a product given that they have certain demographics or purchase history. Understanding conditional probability helps in making informed decisions based on available data.
The probability that a randomly selected person is from Texas, given that they prefer brand A, is approximately 0.47. This is calculated using conditional probability, where we find the number of Texas individuals who prefer brand A divided by the total preferring brand A. The correct choice is option C.
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