91Ó°ÊÓ

In the context of probability, scenarios are different possible outcomes that can occur in an experiment or a situation. Let’s take the example of the royal family. They will continue to have children until they have a boy or reach a maximum of three children. Each child born has two possible outcomes: being a boy or a girl.
This situation leads to several possible scenarios:

• B: They have a boy first, and the process ends.
• GB: They have a girl first and then a boy, ending the process.
• GGB: They have two girls first, and then a boy, ending the process.
• GGG: They have three girls, which means reaching the limit without having a boy.

Each of these scenarios has a specific probability of occurring, which depends on the outcome of each child born. The sum of all scenario probabilities should equals 1, ensuring that all possible outcomes are accounted for.

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. Within our problem, we can form a probability distribution of scenarios for how the royal family might have children.
The probability for having at least one boy or up to three girls can be calculated as:

• For the scenario B: The likelihood of this event occurring is \( P(B) = \frac{1}{2} \).
• For the scenario GB: The probability is determined by having a girl followed by a boy, which is\( P(GB) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).
• For the scenario GGB: Similarly, this probability comes from the sequence of two girls followed by a boy, \( P(GGB) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).
• Lastly, for GGG: The probability is derived from the sequence of three girls, \( P(GGG) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).

These probabilities highlight how likely it is for each scenario to occur, giving us a complete picture of potential family structures under the given conditions.

The expected number of outcomes in probability is an integral part of determining the average result of a random experiment over many trials. To find the expected numbers of boys and girls in our scenario, we calculate these averages using the formula of expected value.
For the expected number of boys, we use:\[ E(B) = 1 \times \frac{1}{2} + 1 \times \frac{1}{4} + 1 \times \frac{1}{8} + 0 \times \frac{1}{8} = \frac{7}{8}. \]This gives us the likelihood-weighted average of boy outcomes in all possible scenarios.
Similarly, for the expected number of girls, we calculate:\[ E(G) = 0 \times \frac{1}{2} + 1 \times \frac{1}{4} + 2 \times \frac{1}{8} + 3 \times \frac{1}{8} = \frac{11}{8}. \]This computation shows the average number of girls expected across all scenarios. By understanding expected values, students can predict the average outcomes from several trials of such random processes, expecting these values to be the most common results over a large number of simulations.