91Ó°ÊÓ

When dealing with probability, understanding conditional probability is crucial. Conditional probability is focused on finding the probability of an event occurring, given that another event has already occurred. In simpler terms, it investigates how the chance of one event is influenced by the occurrence of a different event.

In the archer problem, we are interested in finding how the probability of hitting the target changes when it is windy compared to when it is not. Mathematically, conditional probability is expressed as:

• \( P(A|B) \): the probability of event \( A \) occurring given that \( B \) has occurred.
• This can be calculated using the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

In the solution provided, the conditional probabilities were defined as \( P(H|W) = 0.4 \) for hitting the target in windy conditions and \( P(H|NW) = 0.7 \) for non-windy conditions. Each probability tells us how likely it is to hit the target considering if there is supposed wind or not.

Bayes' theorem is a powerful tool in probability theory that allows one to update the probability measures of an event based on new evidence. It helps us reason backwards from observed events to deduce the probabilistic causes.

In the given problem, we used Bayes’ theorem to find the probability that there was no wind given that the target was missed, represented mathematically as \( P(NW|M) \). This theorem is pivotal when we need to "invert" the conditional probability using the known likelihoods:

• The general form is: \[ P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)} \]

For the archer scenario:

• \( P(NW|M) \) uses the precomputed probability of missing and whether that miss occurred during no wind. It provided insights into the likelihood of conditions based on the outcome.

The concept of joint probability refers to the likelihood of two events occurring simultaneously. This tells us about the intersection of two events, often denoted as \( P(A \cap B) \).

In the archer’s case, step one sought the probability of both windy conditions occurring and hitting the target at the same time, expressed as \( P(W \cap H) \). This joint probability is calculated by multiplying the conditional probability of hitting the target in wind by the occurrence probability of wind:

• \( P(W \cap H) = P(H|W) \cdot P(W) = 0.4 \times 0.3 = 0.12 \)

Understanding joint probability helps in finding situations where multiple criteria must be met simultaneously, which is crucial in probabilistic models.

The rule of total probability is an essential concept that provides a way to calculate the probability of an event by considering all possible scenarios or paths leading up to that event.

This rule is incredibly useful when you are dealing with a partitioned sample space and want to find the overall probability of an event across all these partitions.

In this exercise, calculating the probability that the archer hits the target on her first shot, considering both possibilities of wind presence and absence, involves applying the total probability concept:

• The two separate scenarios (windy and not windy) are considered.
• The individual probabilities from each scenario are summed to find the total probability of the target being hit.
• The calculation was: \( P(H) = P(W \cap H) + P(NW \cap H) = 0.12 + 0.49 = 0.61 \)

By dissecting the total possibility into these distinct yet exhaustive segments, one can arrive at an overall measure of outcome effectively.