Problem 22 - Algebra and Trigonometry

Question

Genders of Children Assume that for any given live human birth, the chances that the child is a boy or a girl are equally likely. (a) What is the probability that in a family of five children a majority are boys? (b) What is the probability that in a family of seven children a majority are girls?

Step-by-Step Solution

Answer

(a) 0.5, (b) 0.5

Step 1: Understand the Problem

We have two scenarios to analyze: (a) a family of five children where a majority are boys and (b) a family of seven children where a majority are girls. The probability of any child being a boy or a girl is 0.5.

Step 2: Define a Binomial Situation

This is a binomial probability scenario since each child can be independently classified as a 'boy' or a 'girl'. For part (a), we need the probability that more than half of the children out of five are boys (i.e., 3, 4, or 5 boys). For part (b), we need the probability that more than half of the children out of seven are girls (i.e., 4, 5, 6, or 7 girls).

Step 3: Calculate Probability for Part (a)

For five children, calculate the probability of getting 3, 4, or 5 boys using the binomial distribution formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n = 5 \), \( k \) is the number of boys, and \( p = 0.5 \).Calculate: - \( P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2 = 10 \cdot 0.125 = 0.3125 \)- \( P(X=4) = \binom{5}{4} (0.5)^4 (0.5)^1 = 5 \cdot 0.0625 = 0.15625 \)- \( P(X=5) = \binom{5}{5} (0.5)^5 (0.5)^0 = 1 \cdot 0.03125 = 0.03125 \)Add these probabilities to find the total probability.

Step 4: Sum Probability for Part (a)

The probability that a family of five children has a majority (3, 4, or 5) of boys is the sum: \[ P(X \geq 3) = 0.3125 + 0.15625 + 0.03125 = 0.5 \].

Step 5: Calculate Probability for Part (b)

For seven children, calculate the probability of getting 4, 5, 6, or 7 girls using the binomial distribution formula with \( p = 0.5 \):- \( P(Y=4) = \binom{7}{4} (0.5)^4 (0.5)^3 = 35 \cdot 0.0625 = 0.2734375 \)- \( P(Y=5) = \binom{7}{5} (0.5)^5 (0.5)^2 = 21 \cdot 0.03125 = 0.1640625 \)- \( P(Y=6) = \binom{7}{6} (0.5)^6 (0.5)^1 = 7 \cdot 0.015625 = 0.0546875 \)- \( P(Y=7) = \binom{7}{7} (0.5)^7 (0.5)^0 = 1 \cdot 0.0078125 = 0.0078125 \)Add these probabilities to find the total probability.

Step 6: Sum Probability for Part (b)

The probability that a family of seven children has a majority (4, 5, 6, or 7) of girls is the sum: \[ P(Y \geq 4) = 0.2734375 + 0.1640625 + 0.0546875 + 0.0078125 = 0.5 \].

Key Concepts

Probability Calculation

Probability calculation is an essential concept that helps us determine the chance of a specific event occurring. It is expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. In this particular scenario, calculating probabilities involve understanding independent events, which is when the outcome of one event does not affect another.
To successfully calculate the probability of multiple children being boys or girls, you apply the rules of probability to each child separately. Since the probability of having either a boy or a girl is 0.5, we can use these basic probabilities in combination with the binomial distribution formula.
Calculations involve adding up the probabilities of outcomes that meet our condition, as seen in the exercises where we needed a majority of boys or girls in a family. This involves using numbers of successful outcomes over all possible outcomes for the event.

Binomial Distribution

Binomial distribution is a statistical method used to model the number of successes in a fixed number of independent trials. Each trial must result in one of two outcomes. For example, in our problem, each child's gender is a trial, and the results are either 'boy' or 'girl'.
The formula for binomial probability is \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:

• \( \binom{n}{k} \) is the number of combinations for selecting \( k \) successes in \( n \) trials.
• \( p \) is the probability of success on each trial.
• \( X=k \) is the event of interest, such as getting \( k \) boys.

For example, finding the probability of a majority of boys involves summing up probabilities for 3, 4, or 5 boys. Similarly, for a majority of girls, we calculate for 4 to 7 girls in seven trials.

Independent Events

Understanding independent events is crucial in statistics. An event is independent if the outcome of one event doesn't influence the outcome of another. In our child-gender scenario, each child's gender is independent of their siblings'.
This independence simplifies probability calculations, as it allows each event to have the same probability, unaffected by the preceding events.
In a sequence of flips for a fair coin, each flip is independent, with a 50% chance for either heads or tails, just like the 50% chance for each child's gender in our problem. When dealing with a series of independent events like this, the same probability calculations apply to each event. This results in using binomial distribution to handle multiple independent trials efficiently.