Problem 5 - Discrete Mathematics with Applications

Question

The Sealords have three children. Assuming that the outcomes are equally likely and independent, find the probability that they have three boys, knowing that: The first child is a boy.

Step-by-Step Solution

Answer

The probability that the Sealords will have three boys, given that their first child is a boy, is \(\boxed{1/4}\) or \(25\%\).

Step 1: Identify known information

We know that: 1. The outcomes are equally likely (with a 50% chance of having a boy and a 50% chance of having a girl). 2. The outcomes are independent (meaning that having a boy or a girl does not affect the chances of having a boy or girl for their other children). 3. The first child is a boy.

Step 2: Calculate the probability of having three boys

Since the probability of having a boy is 50% or \(1/2\). Given that the outcomes are independent and the first child is already a boy, we can simply multiply the probabilities for the remaining two children: Probability of second child being a boy: \(1/2\) Probability of third child being a boy: \(1/2\) Total probability of having three boys: \((1/2) \times (1/2) = (1/4)\)

Step 3: Present the final answer

The probability that the Sealords will have three boys, given that their first child is a boy, is \(\boxed{1/4}\) or \(25\%\).

Key Concepts

Independent Events

When we talk about independent events in probability, we mean situations where the occurrence of one event does not influence the likelihood of another event happening.
This concept is crucial in calculating probabilities, especially in scenarios involving multiple events, like in the case of the Sealords' children.
For the Sealords, the outcome of each child's gender is an independent event since each child has an equal chance of being a boy or a girl, regardless of what happens with the other children.

• The probability of the first child being a boy or a girl does not change the probability for the second or third child.
• This means that the probability of the second child being a boy is always 50% or \(\frac{1}{2}\), the same as for the third child.

Understanding independent events helps simplify the calculations because you can consider each child separately, without factoring in previous outcomes.

Probability Theory

Probability theory is the mathematical framework that helps us quantify the uncertainty and likelihood of various events occurring.
It is essential in many fields, including statistics, finance, science, and everyday life decisions.

• In probability theory, each potential result or outcome has a likelihood measured between 0 and 1. A probability of 1 means an event is certain to happen, while 0 indicates it will not happen.
• Probabilities can also be expressed in percentages, with 100% representing certainty.
• When dealing with multiple independent events, the overall probability of multiple outcomes occurring together is found by multiplying their individual probabilities.

For the Sealords, knowing that the events are independent, we multiply the probability \(\frac{1}{2}\) for the second and third child both being boys to find the overall probability of having three boys given the first is already a boy. Probability theory allows for this straightforward computation.

Sample Space

In probability theory, a sample space is a complete set of all possible outcomes of an experiment or scenario.
Consider it as the universe of potential results that can occur.

• For example, if a family has one child, the sample space for the child's gender is \(\{\text{Boy, Girl}\}\).
• With two children, the sample space expands: \(\{\text{Boy-Boy, Boy-Girl, Girl-Boy, Girl-Girl}\}\).
• In the given problem with the Sealords, since the gender of the first child is known, the sample space for two more children can be simplified to \(\{\text{Boy-Boy, Boy-Girl, Girl-Boy, Girl-Girl}\}\).

By focusing on the relevant part of the sample space — where the first child is a boy — the problem can be simplified. This emphasizes how sample spaces are helpful in structuring problems to enable easier calculation of probabilities.