91Ó°ÊÓ

A binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent experiments. Each experiment, or trial, has two possible outcomes: success or failure.
In our exercise, the success is defined as a child being a girl, and the number of trials is six, since the family has six children. The probability of success (a girl being born) is 0.485, and the probability of failure (a boy being born) is 0.515.
The binomial distribution formula we use looks like this:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Here, \( n \) is the total number of trials, \( k \) is the number of successes, \( p \) is the probability of success, and \( 1-p \) is the probability of failure.
Using this formula, we can compute the probability for any number of girls in the family.

Probability calculation involves determining the likelihood of specific outcomes. To find the probability of certain numbers of girls in a family using the binomial distribution, we use the combination formula and the success/failure probabilities.
For example, calculating for all girls (no boys) in the family of six:
\[ P(X=6) = \binom{6}{6} (0.485)^6 (0.515)^0 \]
This formula translates to:
\[ 1 \cdot 0.485^6 = 0.0114 \]
Similarly, we can follow the same steps for other cases, such as for at least 5 girls. We sum up the probabilities for having exactly 5 girls and exactly 6 girls:
\[ P(X \geq 5) = P(X=5) + P(X=6) \]
Each probability calculation builds on this structure.

Combinatorial analysis is a key part of calculating binomial probabilities. It involves the use of combinations, which are selections of items from a larger set without considering the order.
The combination formula is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
In the context of our exercise, this formula helps to determine the number of ways to choose \( k \) girls out of \( n \) children. For instance, if we need the number of ways to have 5 girls out of 6 children, it's calculated as:
\[ \binom{6}{5} = \frac{6!}{5!(6-5)!} = 6 \]
Combinatorial terms are then multiplied by the success and failure probabilities to get binomial probabilities.
Without combinatorial analysis, determining these probabilities for various outcomes would be difficult.

Probability theory underpins all our calculations through the concepts of random experiments, outcomes, and events.
A core principle is that the sum of probabilities of all possible outcomes must equal 1. For example, in our problem, we're assured that:
\[ P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) = 1 \]
Probability theory also ensures that we can calculate complement events easily. For instance, to find the probability of 'at least 3 girls', we can sum up the probabilities from 3 to 6 directly, as shown:
\[ P(X \geq 3) = P(X=3) + P(X=4) + P(X=5) + P(X=6) \]
It simplifies complex probability calculations, making it easier to deal with real-world scenarios.