Let’s solve each part step-by-step:

Given:

• Probability that $A$ hits the target ($P(A)$) = $\frac{1}{4}$
• Probability that $A$ does not hit the target ($P(A')$) = $1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}$
• Probability that $B$ hits the target ($P(B)$) = $\frac{2}{5}$
• Probability that $B$ does not hit the target ($P(B')$) = $1 - P(B) = 1 - \frac{2}{5} = \frac{3}{5}$

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Part (i): Probability that $A$ does not hit the target

$P(A') = \frac{3}{4}$

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Part (ii): Probability that both $A$ and $B$ hit the target

To find the probability that both hit the target, we use the multiplication rule for independent events: $P(A \cap B) = P(A) \times P(B) = \frac{1}{4} \times \frac{2}{5} = \frac{2}{20} = \frac{1}{10}$

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Part (iii): Probability that one of them hits the target

This occurs in the following two mutually exclusive cases:

1. $A$ hits the target and $B$ does not hit the target: $P(A \cap B') = P(A) \times P(B')$
2. $B$ hits the target and $A$ does not hit the target: $P(A' \cap B) = P(A') \times P(B)$

Adding these probabilities: $P(\text{one of them hits}) = P(A \cap B') + P(A' \cap B)$ Substitute the values: $P(A \cap B') = P(A) \times P(B') = \frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$ $P(A' \cap B) = P(A') \times P(B) = \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$ $P(\text{one of them hits}) = \frac{3}{20} + \frac{6}{20} = \frac{9}{20}$

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Part (iv): Probability that neither $A$ nor $B$ hits the target

This happens when both $A$ and $B$ miss the target: $P(A' \cap B') = P(A') \times P(B')$ Substitute the values: $P(A' \cap B') = \frac{3}{4} \times \frac{3}{5} = \frac{9}{20}$

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Final Answers:

(i) $P(A') = \frac{3}{4}$

(ii) $P(A \cap B) = \frac{1}{10}$

(iii) $P(\text{one of them hits}) = \frac{9}{20}$

(iv) $P(A' \cap B') = \frac{9}{20}$

Let me know if you'd like further clarification or additional examples!

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Related Questions:

1. How do you calculate the probability for three independent events?
2. What is the difference between mutually exclusive and independent events?
3. How do you find the probability of at least one event occurring?
4. Can we use complement probabilities to solve these problems faster?
5. How would the probabilities change if $A$ and $B$ were not independent?

Tip:

When working with probabilities, always verify whether the events are independent or mutually exclusive before performing calculations!