91Ó°ÊÓ

Probability is a way of quantifying the likelihood of an event. Here, an event can be anything that happens or is expected to happen. The probability of an event is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means it is certain. To calculate the probability of an event, we use the formula:
\[ P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \]
In many real-world cases, understanding probabilities helps in making informed decisions. For example, knowing the probability of rain can help you decide whether to carry an umbrella.
In our exercise, we are dealing with the probability of genders in births. Given that the probability of a baby being a boy in China is 0.545, we can easily find the probability of a baby being a girl by subtracting the boy's probability from 1. This is because the two probabilities must add up to 1.

The concept of 'at least one' is incredibly useful in probability calculations. Saying you want the probability of 'at least one' event happening means you are interested in the scenarios where one or more of a particular event occurs.
For instance, in our exercise, we want to know the probability that among six births, there is at least one girl.
The trick to finding 'at least one' probability easily is to use the complementary rule. This rule states that:
`P(at least one) = 1 – P(none)`
Here's why we do this: calculating directly for 'at least one' can sometimes be complicated because it involves many different possibilities. On the other hand, calculating the probability of 'none' is usually straightforward, and then we can simply subtract this value from 1 to get our answer.
So, in our scenario, we first calculate the probability of having no girls out of six births and then subtract this from 1.

Binomial probability deals with experiments where there are two possible outcomes – success or failure. In our example, a 'success' is defined as getting a girl, and a 'failure' is getting a boy. The formula for binomial probability is:
\[ P(X=k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \]
Where:

• \binom{n}{k} is the binomial coefficient
• p is the probability of success
• k is the number of successes
• n is the total number of trials

In our problem, we could use this formula to find the probability of getting exactly any number of girls in six births, but since we need 'at least one girl', using the complementary method as described earlier is simpler.
Understanding binomial probability is critical in many fields, including genetics, quality control in manufacturing, and even in predicting sports outcomes. It shows how we can model and calculate the probabilities of outcomes over a series of trials.