"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
21. BRCA1 Gene Research has shown that approximately 1 woman in 400 carries a mutation of the BRCA1 gene. About 64% of women with this mutation develop breast
cancer. Find the probability that a randomly selected woman will carry the mutation of the BRCA1 gene and will develop breast cancer. (Source: National Cancer Institute)

검증된 단계별 안내

Step 1: Understand the problem. The goal is to find the probability that a randomly selected woman both carries the BRCA1 gene mutation and develops breast cancer. This is a joint probability, which can be calculated using the Multiplication Rule.

Step 2: Recall the Multiplication Rule for probabilities. The rule states that the probability of two events A and B occurring together (P(A and B)) is equal to the probability of A occurring (P(A)) multiplied by the probability of B occurring given that A has occurred (P(B|A)).

Step 3: Identify the given probabilities. From the problem, we know: P(A) = Probability of carrying the BRCA1 gene mutation = 1/400, and P(B|A) = Probability of developing breast cancer given the mutation = 64% or 0.64.

Step 4: Apply the Multiplication Rule. Substitute the values into the formula: P(A and B) = P(A) × P(B|A). This will give the probability of a woman carrying the mutation and developing breast cancer.

Step 5: Interpret the result. Once calculated, the result will represent the likelihood of a randomly selected woman both carrying the BRCA1 gene mutation and developing breast cancer. Ensure the units and context are clear.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Multiplication Rule

The Multiplication Rule in probability states that the probability of two independent events occurring together is the product of their individual probabilities. In this context, it helps calculate the likelihood of a woman both carrying the BRCA1 mutation and developing breast cancer by multiplying the probability of each event.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. In this scenario, the probability of developing breast cancer is conditional on having the BRCA1 mutation, which is crucial for accurately determining the overall probability of both events happening.

Sample Space

The sample space is the set of all possible outcomes of a probability experiment. In this case, the Venn diagram illustrates the sample space of women, highlighting those with the BRCA1 mutation and those who develop breast cancer, which aids in visualizing the relationships between these two groups and calculating the desired probabilities.