Births in Vietnam In Vietnam, the probability of a baby being a b...

4. Probability

Complements

Statistics

4. Probability

Complements

2:20 minutes

Problem 4.3.8b

Textbook Question

Births in Vietnam In Vietnam, the probability of a baby being a boy is 0.526 (based on the data available at this writing). For a family having four children, find the following.

b. The probability that all four children are girls.

Verified step by step guidance

Step 1: Understand the problem. The probability of a baby being a girl is the complement of the probability of a baby being a boy. Since the probability of a boy is 0.526, the probability of a girl is 1 - 0.526 = 0.474.

Step 2: Recognize that the problem involves a binomial probability distribution. The binomial probability formula is: P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where n is the number of trials (children), k is the number of successes (girls), and p is the probability of success (girl).

Step 3: For this problem, we are finding the probability that all four children are girls. This means k = 4 (all successes), n = 4 (four children), and p = 0.474 (probability of a girl).

Step 4: Substitute the values into the binomial probability formula. Since k = n = 4, the combination term (n choose k) simplifies to 1. The formula becomes: P(X = 4) = (1) * (0.474^4) * ((1-0.474)^0).

Step 5: Simplify the expression. The term (1-0.474)^0 equals 1, so the probability simplifies to P(X = 4) = 0.474^4. Calculate this value to find the final probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of a baby being a boy is given as 0.526, which implies that the probability of a baby being a girl is 1 - 0.526 = 0.474. Understanding how to calculate probabilities is essential for solving questions related to outcomes in a given scenario.

Independent Events

Independent events are those whose outcomes do not affect each other. In this case, the gender of each child is independent of the others, meaning the probability of each child being a girl remains constant regardless of the genders of the other children. This concept is crucial for calculating the overall probability of multiple events occurring together.

Binomial Probability Formula

The binomial probability formula is used to determine the probability of a specific number of successes in a fixed number of independent trials, given a constant probability of success. For this question, we can use the formula to find the probability of having all four children as girls, which involves raising the probability of having a girl to the power of the number of children (0.474^4). This formula is fundamental in scenarios involving multiple trials with two possible outcomes.

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Births in Vietnam In Vietnam, the probability of a baby being a boy is 0.526 (based on the data available at this writing). For a family having four children, find the following.b. The probability that all four children are girls.