91Ó°ÊÓ

The mean, often referred to as the average, gives us an idea of the central value of a set of numbers. In statistics, the mean is calculated by adding all of the numbers in a dataset together and then dividing by the number of numbers in the set. Here, for the problem involving 8 births, we determined the mean using the formula \(\text{mean} = \text{n} \times \text{p}\). Since the probability \(\text{p}\) of having a girl is typically 0.5, and there are 8 children (\(\text{n} = 8\)), the mean number of girls among these children is calculated as follows: \[\text{mean} = 8 \times 0.5 = 4\ \]. This means, on average, we expect to have 4 girls in 8 births.

The standard deviation measures the amount of variation or dispersion in a dataset. In simple terms, it tells us how spread out the values in a dataset are around the mean. A low standard deviation means that the values are close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. For our exercise, we calculate the standard deviation using the formula \( \text{standard deviation} = \sqrt{ \text{n} \times \text{p} \times (1 - \text{p})} \). Plugging in our values, we have \[\text{standard deviation} = \sqrt{8 \times 0.5 \times 0.5} = \sqrt{2} \approx 1.414 \]. This tells us that the number of girls in a subset of 8 births can vary by approximately 1.414 around the mean of 4.

A binomial distribution is a discrete probability distribution that summarizes the likelihood that a value will take one of two independent states across a series of observations. This is applicable when there are two possible outcomes, such as success and failure. In our scenario, the two outcomes are having a girl or not having a girl (which we'll call having a boy), with the probability \(\text{p} = 0.5\) for each birth. The binomial distribution formula helps us determine the probability of achieving a fixed number of successes (girls in this case) in a specified number of trials (births). We can summarize the key properties of a binomial distribution as:

• Fixed number of trials (n).
• Two possible outcomes for each trial.
• A constant probability of success (p) for each trial.
• Independent trials where previous outcomes don't affect future ones.

In our problem, we use the provided probabilities from the table to analyze the outcomes in groups of 8 births.

Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event). We use probability to assess the chance of various outcomes in statistical experiments. For example, the probability of having a girl in a single birth is \(\text{p} = 0.5\). Probability helps us make predictions about future events based on known data. In the context of our problem, the probability (\tabs{P}{x}}) for each possible number of girls (\(\text{x})\), such as 0, 1, 2,...,8, has been provided. Using these probabilities, we can assess how likely it is to have, for example, exactly 6 girls in 8 births. By comparing the probability of 6 girls with other values, we observed that while having 6 girls is unusual, it is not significantly high because it falls within the expected range defined by our calculation of mean and standard deviation.