Problem 68 - Precalculus with Limits

Question

Assume that the probability of the birth of a child of a particular sex is \( 50\% \). In a family with four children, what are the probabilities that(a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Step-by-Step Solution

Answer

The probabilities are: (a) 6.25%, (b) 12.5%, (c) 93.75%

Step 1: Calculate the probability of all children being boys

For each child, the probability of being a boy is \( 0.5 \) (50%). Since there are 4 children, and they are independent events, the probability of all children being boys is \( 0.5^4 = 0.0625 \) (6.25%).

Step 2: Calculate the probability of all children being the same sex

Here we have two independent events to consider: all children being boys and all children being girls. As we've calculated in step 1, the probability of all children being boys is \( 0.0625 \). Similarly, the probability of all children being girls also is \( 0.5^4 = 0.0625 \). Combining both possibilities gives \( 2 * 0.0625 = 0.125 \) (12.5%).

Step 3: Calculate the probability of there being at least one boy

To find this probability, it's easier to calculate the complementary event: that is, all children are girls. As calculated in step 1, this is \( 0.0625 \). So, the probability of there being at least one boy is \( 1 - 0.0625 = 0.9375 \) (93.75%).

Key Concepts

Independent Events

When studying probability in the context of familial traits, like the sex of children born to a family, the concept of independent events is essential. An independent event is one where the outcome of one event does not influence the outcome of another. In our exercise example, each child being born as a boy or a girl is an independent event. The probability of having a boy (50) doesn't change regardless of the sex of previous children. The concept is crucial when calculating probabilities for multiple children, as one would multiply the individual probabilities to get the combined result. For instance, the probability of having four boys in a row would be calculated as (0.5)^4. Understanding independent events helps to break down complex probability questions into simpler, manageable calculations.

Complementary Events

With complementary events, we're dealing with two outcomes that together cover all possible scenarios. These events are mutually exclusive and exhaustive—meaning they cannot occur at the same time, yet one of them must occur. In probability theory, the probability of an event happening plus the probability of its complement must equal 1. This principle is beautifully illustrated in our example where the probability of at least one boy being born is the complement of the probability of no boys (all girls) being born. Given that the probability of all girls is 0.0625, the complement and thus the probability of at least one boy is 1 - 0.0625 = 0.9375, leveraging the rule that the probabilities of complementary events always sum to 1.

Probability Calculations

The backbone of analyzing scenarios like those in our family example lies in solid probability calculations. It involves an understanding of basic probability rules and applying them correctly. In the given scenario, calculating the probability for various sex distributions among four children requires a mix of multiplication for independent events and addition or subtraction for joint or complementary probabilities. For instance, the probability of all four children being the same sex involves adding the probability of all boys to the probability of all girls (2 * 0.0625). Knowing how to approach each situation—whether it requires the multiplication rule for independent events, the addition rule for non-overlapping events, or the complementary rule for opposite scenarios—is key to accurate and effective probability calculations.