Bayesian Probability Application: Target Shooting Scenario

Suppose we have two shooters, A and B, with shooting probabilities of 0.4 and 0.7, respectively. This problem involves determining the probability that Shooter A hits the target given that the target is hit at least once. We'll use Bayesian probability to solve this.

Understanding the Problem:

Let's define the events:

The probability that Shooter A hits the target is PA 0.4. The probability that Shooter B hits the target is PB 0.7.

Calculating the Probability of the Target Being Hit:

To find the probability that the target is hit at least once, we need to consider the complementary probability, where the target is missed by both shooters.

Step 1: Calculate the Miss Probabilities

The probability that Shooter A misses the target is Pmiss A 1 - PA 0.6.

The probability that Shooter B misses the target is Pmiss B 1 - PB 0.3.

Step 2: Calculate the Probability of Both Missing the Target

The probability that both shooters miss the target is:

Pmiss both Pmiss A times; Pmiss B 0.6 times; 0.3 0.18

Therefore, the probability that the target is hit at least once is:

Phit at least once 1 - Pmiss both 1 - 0.18 0.82

Applying Bayes' Theorem to Find the Conditional Probability

We use Bayes' Theorem to find the probability that Shooter A hits the target given that the target is hit at least once.

The formula for Bayes' Theorem in this context is:

P(A|hit at least once) (P(hit at least once | A hits) times; PA) / P(hit at least once)

Step 3: Calculate the Joint Probability

If Shooter A hits the target, the probability that the target is hit at least once is 1, since Shooter B could either hit or miss. Hence,

P(hit at least once | A hits) 1

Plugging the values into Bayes' Theorem, we get:

P(A hits | hit at least once) (1 times; 0.4) / 0.82 0.4 / 0.82 ≈ 0.4878

The probability that Shooter A hits the target given that the target is hit at least once is approximately 0.4878, or 48.78%.

Counting the Hits:

We can also use a more direct method to calculate the probability that the target is hit once.

To be found is:

PAE PA ∩ E / PE, where E is the event that the target is hit once.

Step 4: Calculate PA ∩ E

The probability that Shooter A hits and Shooter B misses is:

PA ∩ Bc PA times; PBc 0.4 times; 0.3 0.12

Step 5: Calculate PE

The probability that either Shooter A hits and Shooter B misses or Shooter B hits and Shooter A misses is:

PA ∩ Bc PAc ∩ B 0.4 times; 0.3 0.6 times; 0.7 0.12 0.42 0.54

Finally, the probability that Shooter A hits the target given that the target is hit once is:

PAE (0.12) / (0.54) 2/9 ≈ 0.2222

Conclusion:

From the above calculations, we find that the probability that Shooter A hits the target given that the target is hit at least once can be either approximately 0.4878 or 0.2222, depending on the specific probability model used. Both methods confirm the use of Bayesian probability to solve such conditional probability problems.