Problem 22 - Technical Mathematics with Calculus

Question

If male and female births are equally likely, what is the probability of five births being all girls?

Step-by-Step Solution

Answer

The probability of five births being all girls is \(0.5^5 = 0.03125\text{ or }3.125\text{%}\text{.}\)

Step 1: Understanding the Problem

We are given that the probability of a child being born a girl is equal to that of a child being born a boy. This means that each birth can be seen as a binary event with two possible outcomes: girl (G) or boy (B). Since the events are independent, the probability of each event is the same for each birth. The task is to find the probability of having five girls (GGGGG) in a row.

Step 2: Calculating the Probability of One Girl

Since births are equally likely to be male or female, the probability (P) of one child being a girl is 0.5 or 50%.

Step 3: Calculating the Probability of Five Girls

The births are independent events, so the probability of five girls in a row is the product of the probability of each event happening. Therefore, we multiply the probability of one girl by itself five times: P(GGGGG) = P(G) * P(G) * P(G) * P(G) * P(G).

Key Concepts

Binary Events in Probability

When discussing probability, the notion of binary events is fundamental. These are situations with two mutually exclusive outcomes; for example, flipping a coin results in either 'heads' or 'tails.' In the context of births, as given in our exercise, we define the sex of a newborn as a binary event - it can be a girl (G) or a boy (B), but not both.

Understanding binary events is crucial for students as it sets the foundation for more complex probability scenarios. In the given exercise, with each birth being independent and having a binary outcome, we’re looking at repeated instances of a binary event. This simplicity allows us to use straightforward multiplication to calculate the combined probability of successive events.

Independent Events in Probability

The concept of independent events is another key pillar of probability theory. In the realm of probability, events are considered independent if the outcome of one event does not influence the outcome of another. This means that the result of one birth being a boy or girl does not affect the result of the next birth.

To illustrate, imagine rolling a die. The result of the first roll does not affect the result of the second roll - each roll is independent of the other. Similarly, in our exercise problem, the probability of each birth resulting in a girl is not impacted by previous births. This independency is why we can multiply the probabilities of each event together when calculating the overall likelihood of consecutive outcomes.

Calculating Probabilities

The process of calculating probabilities may seem daunting at first, but once the principles of binary and independent events are understood, it becomes much more manageable. To calculate the probability of multiple independent events occurring in sequence, as in the birth scenario provided, one must simply multiply the individual probabilities of each event.

In the exercise, the probability of a single birth resulting in a girl is 0.5. Since the births are independent, the probability of five girls is calculated by multiplying the probability of one girl five times, represented mathematically as \(P(GGGGG) = P(G) \times P(G) \times P(G) \times P(G) \times P(G)\). This results in \(0.5^5\), or 0.03125, which is a 3.125% chance of all five births being girls. Understand that this calculation method holds true as long as the binary events are independent and have consistent probability outcomes.