1. **Problem statement:** We have a pack of 10 cards: 6 red cards numbered 1, 2, 3, 4, 5, 6 and 4 black cards numbered 1, 2, 3, 4. We analyze two events for part (a): - $R$: the card drawn is red - $E$: the card drawn has an even number We want to check if $R$ and $E$ are independent. 2. **Recall the independence rule:** Two events $A$ and $B$ are independent if and only if $$P(A \cap B) = P(A) \times P(B)$$ 3. **Calculate probabilities:** - Total cards: 10 - $P(R) = \frac{6}{10} = 0.6$ - Even numbers are 2, 4, 6. Cards with even numbers: - Red even numbers: 2, 4, 6 (3 cards) - Black even numbers: 2, 4 (2 cards) - Total even cards: 5 - $P(E) = \frac{5}{10} = 0.5$ 4. **Calculate $P(R \cap E)$:** Cards that are both red and even are 2, 4, 6 in red, so 3 cards. $$P(R \cap E) = \frac{3}{10} = 0.3$$ 5. **Check independence:** Calculate $P(R) \times P(E) = 0.6 \times 0.5 = 0.3$ Since $$P(R \cap E) = P(R) \times P(E) = 0.3,$$ $R$ and $E$ are independent. 6. **Part (b) problem statement:** Events: - $B$: the card is black - $T$: the card number is 2 7. **Calculate probabilities:** - $P(B) = \frac{4}{10} = 0.4$ - Cards with number 2: red 2 and black 2, total 2 cards - $P(T) = \frac{2}{10} = 0.2$ 8. **Calculate $P(B \cap T)$:** Cards that are black and number 2: only black 2 card, so 1 card. $$P(B \cap T) = \frac{1}{10} = 0.1$$ 9. **Check independence:** Calculate $P(B) \times P(T) = 0.4 \times 0.2 = 0.08$ Since $$P(B \cap T) = 0.1 \neq 0.08 = P(B) \times P(T),$$ $B$ and $T$ are not independent. **Final answers:** - (a) $R$ and $E$ are independent because $P(R \cap E) = P(R)P(E)$. - (b) $B$ and $T$ are not independent because $P(B \cap T) \neq P(B)P(T)$.