To solve the problems step by step, we will first address question 2.2 regarding the probabilities of events A and B.
2.2.1 Calculate P(A and B)
We can use the formula for the probability of the union of two events:
\[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \]
Given:
- \( P(A) = 0.3 \)
- \( P(B) = 0.4 \)
- \( P(A \text{ or } B) = 0.7 \)
Plugging in the values we have:
\[ 0.7 = 0.3 + 0.4 - P(A \text{ and } B) \]
Now, solve for \( P(A \text{ and } B) \):
\[ 0.7 = 0.7 - P(A \text{ and } B) \]
Rearranging gives:
\[ P(A \text{ and } B) = 0.7 - 0.7 = 0 \]
So, we find:
\[ P(A \text{ and } B) = 0 \]
2.2.2 Are events A and B independent? Show your calculations.
Two events A and B are considered independent if:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
From our previous calculation, we found \( P(A \text{ and } B) = 0 \).
Now, we calculate \( P(A) \times P(B) \):
\[ P(A) \times P(B) = 0.3 \times 0.4 = 0.12 \]
Since \( P(A \text{ and } B) \neq P(A) \times P(B) \) (0 is not equal to 0.12), events A and B are not independent.
Summary of 2.2
- \( P(A \text{ and } B) = 0 \)
- Events A and B are not independent since \( P(A \text{ and } B) \neq P(A) \times P(B) \).
2.3 Classify the following events:
2.3.1 Rolling an odd number on a die and rolling an even number on a die.
- Mutually Exclusive: Two events are mutually exclusive if they cannot happen at the same time. Since you cannot roll an odd number and an even number on a single roll of a die at the same time, these events are mutually exclusive.
Classification: Mutually Exclusive
2.3.2 Rain falling in South Africa and winning the lottery in Spain.
- Independent: Events are independent if the occurrence of one does not affect the occurrence of the other. Rain in South Africa and winning the lottery in Spain are unrelated events; the outcome of one does not impact the other.
Classification: Independent
Summary of 2.3
- 2.3.1: Mutually Exclusive
- 2.3.2: Independent