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Worked Examples: Interpreting a Binomial Distribution

Assume the probability of a boy birth and a girl birth is equally likely. The variable [tex]$X$[/tex] represents the number of boys born to a family. A family has three children.

Create a probability distribution table.
\begin{tabular}{|l|c|c|c|c|}
\hline
[tex]$X$[/tex] & 0 & 1 & 2 & 3 \\
\hline
[tex]$P$[/tex] & [tex]$0.125$[/tex] & [tex]$0.375$[/tex] & [tex]$0.375$[/tex] & [tex]$0.125$[/tex] \\
\hline
\end{tabular}

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The possible outcomes are:
BBB, BBG, BGB, BGG, GGG, GGB, GBG, GBB

There is 1 way to have 0 boys, 3 ways to have 1 boy, 3 ways to have 2 boys, and 1 way to have 3 boys.



Answer :

GinnyAnswer
Sure, let's create a probability distribution table for the number of boys born to a family with three children, assuming each child has an equal probability (0.5) of being a boy.

#### Steps:

1. Define the Variables:
- [tex]\( p \)[/tex] = Probability of having a boy = 0.5
- [tex]\( 1 - p \)[/tex] = Probability of having a girl = 0.5
- [tex]\( n \)[/tex] = Number of children = 3

2. Possible Values of X:
- [tex]\( X \)[/tex] can be 0, 1, 2, or 3 (representing the number of boys).

3. Calculate the Probability for Each Value of X using the Binomial Formula:

The formula for a binomial probability [tex]\( P(X = k) \)[/tex] is:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex].

- For [tex]\( X = 0 \)[/tex]:
[tex]\[ P(X = 0) = \binom{3}{0} (0.5)^0 (0.5)^3 = 1 \times 1 \times 0.125 = 0.125 \][/tex]

- For [tex]\( X = 1 \)[/tex]:
[tex]\[ P(X = 1) = \binom{3}{1} (0.5)^1 (0.5)^2 = 3 \times 0.5 \times 0.25 = 0.375 \][/tex]

- For [tex]\( X = 2 \)[/tex]:
[tex]\[ P(X = 2) = \binom{3}{2} (0.5)^2 (0.5)^1 = 3 \times 0.25 \times 0.5 = 0.375 \][/tex]

- For [tex]\( X = 3 \)[/tex]:
[tex]\[ P(X = 3) = \binom{3}{3} (0.5)^3 (0.5)^0 = 1 \times 0.125 \times 1 = 0.125 \][/tex]

4. Create the Probability Distribution Table:

[tex]\[ \begin{array}{|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.125 & 0.375 & 0.375 & 0.125 \\ \hline \end{array} \][/tex]

#### Conclusion:
The probability distribution for the number of boys in a family with three children, where the probability of each child being a boy is 0.5, is as follows:
- There is a 12.5% chance of having 0 boys.
- There is a 37.5% chance of having 1 boy.
- There is a 37.5% chance of having 2 boys.
- There is a 12.5% chance of having 3 boys.