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.
