91Ó°ÊÓ

When we talk about independent events, it's essential to know that the occurrence of one event does not have any impact on the other. In our example, the life expectancies of the man and woman are independent. This means that whether the man is alive in 10 years does not affect the probability of the woman being alive,
and vice versa. Independent events are a crucial concept in probability because they allow us to compute the likelihood of two events occurring together by simply multiplying their individual probabilities.

• If two events, A and B, are independent, then the probability of both occurring, written as \( P(A \text{ and } B) \), is \( P(A) \times P(B) \).
• It helps simplify complex probability scenarios by breaking them down into individual, unrelated probabilities.

In our exercise, since the events are independent, we directly multiply the given probabilities of the man and woman being alive to find their combined probability.

The complement rule in probability helps you find the likelihood of an event not happening. If the probability of an event happening is \( P(A) \), the probability of it not happening, \( P(A^{c}) \), is given by \( 1 - P(A) \). In our context, the probability of the man not being alive in 10 years is \( 1 - 0.74 = 0.26 \).
Similarly, for the woman, it is \( 1 - 0.94 = 0.06 \).

• The complement rule is especially useful when it's easier to calculate the probability of an event happening versus it not happening.
• Using this rule, we can determine the probability of neither the man nor the woman being alive by finding and multiplying the probability of each not being alive.

The complement rule is a powerful tool that simplifies calculations and gives insight into events' opposing outcomes.

The multiplication rule for independent events allows us to find the probability of multiple independent events happening together. Having independent events means the probability of them happening is unaffected by one another. Thus, we multiply their individual probabilities as already touched upon.
This formula looks like \( P(A \text{ and } B) = P(A) \times P(B) \) when A and B are independent.

• This rule was used to find the probability that both will be alive. The calculation is \( 0.74 \times 0.94 = 0.6956 \).
• It can also be inversely used with complements to calculate the likelihood of neither event happening: \( 0.26 \times 0.06 = 0.0156 \), determining that both are not alive.

The multiplication rule is incredibly helpful in various probability problems where independence is a factor. It becomes a tool for simplifying calculations in seemingly complex scenarios.