23 children born. What is the probability that 22 are boys & 1 is a girl?

To determine the probability that 22 children are boys and 1 is a girl, you need to know the probability of having a boy or girl for a single birth. If you assume that the chances of having a boy or girl are equal, then the probability of having a boy is 1/2 and the probability of having a girl is also 1/2.

It's important to note that we assume each birth is independent of the others. In reality, there might be some correlation or bias, but for this exercise, we'll assume independence.

To calculate the probability of having 22 boys and 1 girl, you need to consider the order of the births as well. Imagine that you have 23 slots, each representing a child. Since there are 22 boys and 1 girl, you need to select 22 of the slots for the boys and 1 for the girl.

The number of ways to do this is given by the combination formula, which is denoted by "nCr" or "nCr". For our case, we have 23 slots and we need to choose 22 for the boys, which can be written as "23C22". The combination formula can be calculated as follows:

nCr = n! / (r!(n - r)!)

where n! is the factorial of n, defined as n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1. In our case, n! = 23 * 22 * 21 * ... * 3 * 2 * 1.

Substituting the values into the formula, we get:

23C22 = 23! / (22!(23 - 22)!)
= 23! / (22! * 1!)
= 23 / 1
= 23

So, there are 23 different ways to arrange 22 boys and 1 girl in 23 slots.

Now that we know the number of possible outcomes, we can calculate the probability. Since each outcome is equally likely, the probability is the number of desired outcomes divided by the number of possible outcomes:

Probability = (Number of desired outcomes) / (Number of possible outcomes)
= 1 / 23
= 0.0435 (approximately)

Therefore, the probability that 22 children are boys and 1 is a girl is approximately 0.0435 or 4.35%.