To solve this problem, let’s break it down step by step.

Problem Restatement:

• Person A hits the target 3 times out of 4 shots, so the probability of hitting the target is: $P_A = \frac{3}{4}$
• Person B hits the target 2 times out of 3 shots, so the probability of hitting the target is: $P_B = \frac{2}{3}$
• We are tasked to find the probability that the target is not hit by both A and B.

Key Concept:

The probability of a person not hitting the target is: $P(\text{Not hitting}) = 1 - P(\text{Hitting}).$ We then calculate the probability that both A and B miss the target, which involves multiplying their individual probabilities of missing (assuming independence between A and B).

Step 1: Probability of A not hitting the target

The probability of A not hitting the target is: $P(\text{A misses}) = 1 - P_A = 1 - \frac{3}{4} = \frac{1}{4}.$

Step 2: Probability of B not hitting the target

The probability of B not hitting the target is: $P(\text{B misses}) = 1 - P_B = 1 - \frac{2}{3} = \frac{1}{3}.$

Step 3: Probability that both A and B miss the target

To find the probability that both A and B miss the target, multiply the probabilities of A missing and B missing: $P(\text{Both miss}) = P(\text{A misses}) \times P(\text{B misses}).$ Substitute the values: $P(\text{Both miss}) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}.$

Final Answer:

The probability that the target is not hit by both A and B is: $\boxed{\frac{1}{12}}$

Summary:

1. Person A's probability of missing: $\frac{1}{4}$
2. Person B's probability of missing: $\frac{1}{3}$
3. Combined probability that both miss: $\frac{1}{12}$

Let me know if you need further clarification or have any additional questions! 😊

Here are 5 related questions to expand your understanding:

1. What is the probability that at least one of them hits the target?
2. If Person A takes 5 shots, hitting the target 4 times, what is their hitting probability?
3. How do you calculate combined probabilities for dependent events?
4. What is the probability that both hit the target simultaneously?
5. If the probabilities change, how do we adjust the calculation?

Tip: Always check for independence when combining probabilities of two events!