Problem 50 - A Graphical Approach to College Algebra

Question

(a) Find the probabilities of having \(0,1,2,\) or 3 boys in a family of 3 children. (b) Find the probabilities of having \(0,1,2,3,4,5,\) or 6 girls in a family of 6 children.

Step-by-Step Solution

Answer

Use binomial probability to find: a) P(0)=0.125, P(1)=0.375, P(2)=0.375, P(3)=0.125; b) P(0)=0.0156, P(1)=0.0938, P(2)=0.2344, P(3)=0.3125, P(4)=0.2344, P(5)=0.0938, P(6)=0.0156.

Step 1: Understanding the Problem

In this problem, we need to find the probabilities of a certain number of boys or girls out of given children. Since each child can be either a boy or a girl with equal probability, we use the binomial probability formula.

Step 2: Define the Binomial Probability Formula

The binomial probability formula is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n \) is the total number of trials (children), \( k \) is the number of successes (boys or girls), \( p \) is the probability of success, usually 0.5 for boys or girls.

Step 3: Calculate Probabilities for Boys (Part a)

For three children, the probability of having 0, 1, 2, or 3 boys is calculated as follows:- For 0 boys: \[ P(X = 0) = \binom{3}{0}(0.5)^0 (0.5)^3 = 0.125 \]- For 1 boy: \[ P(X = 1) = \binom{3}{1}(0.5)^1 (0.5)^2 = 0.375 \]- For 2 boys: \[ P(X = 2) = \binom{3}{2}(0.5)^2 (0.5)^1 = 0.375 \]- For 3 boys: \[ P(X = 3) = \binom{3}{3}(0.5)^3 (0.5)^0 = 0.125 \]

Step 4: Calculate Probabilities for Girls (Part b)

For six children, the probability of having 0, 1, 2, 3, 4, 5, or 6 girls is calculated as follows:- For 0 girls: \[ P(X = 0) = \binom{6}{0}(0.5)^0 (0.5)^6 = 0.0156 \]- For 1 girl: \[ P(X = 1) = \binom{6}{1}(0.5)^1 (0.5)^5 = 0.0938 \]- For 2 girls: \[ P(X = 2) = \binom{6}{2}(0.5)^2 (0.5)^4 = 0.2344 \]- For 3 girls: \[ P(X = 3) = \binom{6}{3}(0.5)^3 (0.5)^3 = 0.3125 \]- For 4 girls: \[ P(X = 4) = \binom{6}{4}(0.5)^4 (0.5)^2 = 0.2344 \]- For 5 girls: \[ P(X = 5) = \binom{6}{5}(0.5)^5 (0.5)^1 = 0.0938 \]- For 6 girls: \[ P(X = 6) = \binom{6}{6}(0.5)^6 (0.5)^0 = 0.0156 \]

Key Concepts

Probability

When we talk about probability, we are referring to the chance or likelihood of a particular event occurring. It's a way to quantify randomness and uncertainty. Probability values range between 0 and 1, where 0 means an event is impossible, and 1 indicates certainty. For example, when flipping a fair coin, the probability of landing heads is 0.5, as there are two equally likely outcomes: heads or tails.

Probability is the backbone of many areas in mathematics and science, such as statistics and physics. In our exercise, probability helps us understand how likely it is to have a certain number of boys or girls in a family given some assumptions about birth likelihoods being equal. Remember:

• Probability = Number of desired outcomes / Total number of possible outcomes.
• The sum of probabilities for all possible outcomes of a random event adds up to 1.

Binomial Formula

The binomial formula is crucial when dealing with scenarios where there are exactly two outcomes, often called "success" and "failure." In the context of our exercise, each child can either be a boy (success) or a girl (failure), and we assume these events happen independently with equal probability.

The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:

• \( n \) is the total number of trials or events (children in the family).
• \( k \) is the number of successful outcomes you're interested in (like having 3 boys).
• \( p \) is the probability of a single success (here, 0.5 for either a boy or a girl).

The binomial coefficient \( \binom{n}{k} \) is calculated using factorials:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
It's like picking \( k \) successes in \( n \) trials, showing how many ways we can achieve this combination. When you plug values into this formula, you get the probability of getting \( k \) boys or girls in a given number of children.

Independent Events

Independent events are a fundamental concept in probability theory. Two events are independent if the occurrence of one does not affect the occurrence of the other. This is a key assumption in using the binomial formula.

In our exercise, we assume the birth of each child is independent of the others. This means the probability of one child being a boy is the same regardless of the gender of other children in the family.

• Independence implies that the occurrence of one event provides no information about another.
• For independent events, the probability of both occurring is the product of their probabilities: if \( A \) and \( B \) are two independent events, then \( P(A \text{ and } B) = P(A) \times P(B) \).

Understanding independence is crucial for correctly applying probability models like the binomial distribution. It simplifies calculations and helps predict outcomes accurately in situations like our exercise.