91Ó°ÊÓ

In the world of probability, the concept of independence is a cornerstone. For this exercise, we assume that the events "Barbara hits the target" and "Dianne hits the target" are independent. But what does this really mean? Independence suggests that the result of Barbara's shot does not influence the result of Dianne’s shot. In simpler terms, neither shooter can alter the other's chance success.

This assumption is crucial because it allows us to simplify calculations significantly. When two events are independent, we can directly multiply their probabilities to find the likelihood of both occurring. Without independence, calculations become more complex, requiring a different approach to determine how one event's outcome might affect the other.

Remember: independence is an assumption that is not always true in real-world situations, but it helps us model and solve problems like this in mathematical exercises.

The multiplication rule is a handy tool in probability that tells us how to find the probability of two simultaneous independent events. For shooting scenarios, it helps us determine the probability both Barbara and Dianne hit the target.

When two events, say A (Barbara hits) and B (Dianne hits), are independent, their conjunction, or occurence together, is represented by \( A \cap B \). The multiplication rule states:

• \( P(A \cap B) = P(A) \times P(B) \)

So, if you know the probability of Barbara hitting the duck is \( p_1 \) and Dianne hitting it is \( p_2 \), you simply multiply these probabilities to find the chance they both hit the duck.

• \( P(A \cap B) = p_1 \times p_2 \)

This rule simplifies complex problems into simple multiplication, provided the events are independent. Keep in mind: if events aren't independent, this method doesn't apply.

The addition rule is a fundamental rule that helps you calculate the probability of at least one of several events happening. For our target shooting example, it considers the chance that either Barbara, Dianne, or both, successfully hit the target.

When dealing with independent events A and B, the addition rule states:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Here's how this works for Barbara and Dianne:

• The probability either one of them hits the target is \( p_1 + p_2 \).
• Because their shots are independent, you subtract the probability both hit the target, \( p_1 \times p_2 \), to avoid double-counting.

So, substituting the given probabilities, you calculate:

• \( P(A \cup B) = p_1 + p_2 - p_1 \times p_2 \)

Once you apply this rule, you find the probability of at least one shot hitting the duck. Hence, the addition rule assists you in understanding the true combined probability of independent events without overlaps.