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Question 1: Anna and Rob take their driving tests on the same day. The probability of Anna passing her driving test is 0.7. The probability of both Anna and Rob passing is 0.35.

(a) Work out the probability of Rob passing his driving test.

(b) Work out the probability of both Anna and Rob failing their driving tests.



Answer :

GinnyAnswer
Let's solve this step-by-step:

Given Data:

- Probability of Anna passing (P(A)) = 0.7
- Probability of both Anna and Rob passing (P(A ∩ R)) = 0.35

### Part (a): Probability of Rob passing his driving test

To find the probability of Rob passing (P(R)), we use the formula for the probability of both events happening.

[tex]\[ P(A \cap R) = P(A) \times P(R) \][/tex]

Given [tex]\( P(A \cap R) \)[/tex] and [tex]\( P(A) \)[/tex], we can rearrange the formula to solve for [tex]\( P(R) \)[/tex]:

[tex]\[ P(R) = \frac{P(A \cap R)}{P(A)} \][/tex]

Substituting the given values:

[tex]\[ P(R) = \frac{0.35}{0.7} = 0.5 \][/tex]

So, the probability of Rob passing his driving test is 0.5.

### Part (b): Probability of both Anna and Rob failing their driving tests

First, we need to find the probabilities of Anna and Rob failing individually.

- Probability of Anna failing (P(A')) is the complement of the probability of Anna passing:

[tex]\[ P(A') = 1 - P(A) \][/tex]
[tex]\[ P(A') = 1 - 0.7 = 0.3 \][/tex]

- Probability of Rob failing (P(R')) is the complement of the probability of Rob passing:

[tex]\[ P(R') = 1 - P(R) \][/tex]
[tex]\[ P(R') = 1 - 0.5 = 0.5 \][/tex]

Now, to find the probability of both Anna and Rob failing, we multiply their individual probabilities of failing:

[tex]\[ P(A' \cap R') = P(A') \times P(R') \][/tex]
[tex]\[ P(A' \cap R') = 0.3 \times 0.5 = 0.15 \][/tex]

So, the probability of both Anna and Rob failing their driving tests is 0.15.

Summary:

(a) The probability of Rob passing his driving test is 0.5.

(b) The probability of both Anna and Rob failing their driving tests is 0.15.

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