Problem 62 - A First Course in Probability

Question

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits a wooden duck target with probability \(p_{1},\) while each shot of Dianne's hits it with probability \(p_{2} .\) Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that (a) both shots hit the duck? (b) Barbara's shot hit the duck? What independence assumptions have you made?

Step-by-Step Solution

Answer

(a) The probability that both shots hit the duck is given by: \[P(A \cap B) = p_1 \times p_2\] (b) The probability that Barbara's shot hit the duck if the duck was knocked over is given by: \[P(A | A \cup B) = \frac{p_1}{p_1 + p_2 - p_1 \times p_2}\] We assume that both Barbara's and Dianne's shots are independent events, meaning the outcome of one shot does not affect the outcome of the other shot.

Step 1: Understand the problem and notation

We are given the probability of Barbara and Dianne hitting the target, denoted by \(p_1\) and \(p_2\), respectively. We need to find the probability that both shots hit the duck and the probability that Barbara's shot hit the duck if the duck was knocked over. Denote the events as follows: - A: Barbara hits the target - B: Dianne hits the target We are looking for the probabilities of A ∩ B (both hit the target) and A, given that the target was hit (A ∩ B or A ∪ B).

Step 2: Calculate the probability of both shots hitting the target

To find the probability of both shots hitting the target (A ∩ B), we need to use the multiplication rule. If events A and B are independent, the probability that both A and B happen is given by: \[P(A \cap B) = P(A) \times P(B)\] Since we are assuming their shots are independent, we can calculate the probability of both hitting the target as: \[P(A \cap B) = p_1 \times p_2\]

Step 3: Calculate the probability of either shot hitting the target

Since A and B are independent events, we can find the probability of at least one of them hitting the target (A ∪ B) using the addition rule: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Substituting the given probabilities and the probability from Step 2, we get: \[P(A \cup B) = p_1 + p_2 - p_1 \times p_2\]

Step 4: Calculate the conditional probability that Barbara's shot hit the target, given the duck was knocked over

To find the probability that Barbara's shot hit the duck if the duck was knocked over (A ∣ A ∪ B), we use the conditional probability formula: \[P(A | A \cup B) = \frac{P(A \cap (A \cup B))}{P(A \cup B)}\] Since \(A \cap (A \cup B) = A\), the formula becomes: \[P(A | A \cup B) = \frac{P(A)}{P(A \cup B)}\] Now, substitute the given probabilities and the result from Step 3: \[P(A | A \cup B) = \frac{p_1}{p_1 + p_2 - p_1 \times p_2}\] The independence assumptions we have made are that both Barbara's and Dianne's shots are independent events, i.e., the outcome of one shot does not affect the outcome of the other shot.

Key Concepts

Independence Assumptions

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.

Multiplication Rule

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.

Addition Rule

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.