(a) Find the conditional probability that the black die resulted in a number less than 4 given (i) P(E ∩ F) and P(F) (ii) P(E ∪ F) and P(E, F) (iii) P(E) and P(G | E).

Steps to Solve

1. Understanding Given Events

   We have to find various probabilities based on conditions related to rolling two dice. Let:

   • E: "The event that the sum is greater than 9"
   • F: "The event that the black die is rolled"
   • G: "The event that the red die is rolled"

2. Calculate Total Outcomes

   The total possible outcomes when rolling two six-sided dice is given by: $$ \text{Total Outcomes} = 6 \times 6 = 36 $$

3. Finding Event P(E)

   Calculate the number of outcomes where the sum of two dice is greater than 9.

   • The combinations for sums greater than 9:
     • Sum = 10: (4,6), (5,5), (6,4)
     • Sum = 11: (5,6), (6,5)
     • Sum = 12: (6,6)
   • Total outcomes where the sum is greater than 9 = 6

   Thus, $$ P(E) = \frac{6}{36} = \frac{1}{6} $$

4. Defining Condition P(F) and P(G)

   Assume F and G represent specific outcomes from one of the dice:

   • F = black die lands on a specific number
   • G = red die lands on a specific number

5. Finding P(F and E)

   Determine the joint probability of the events F and E:

   • You need to consider which combinations from F contribute to E.
   • For example, if F = 5, the combinations that yield a sum greater than 9 would be (5,5), (5,6).

6. Conditional Probability Calculation

   Calculate the conditional probabilities:

   • The conditional probability of G given E would be: $$ P(G|E) = \frac{P(G \cap E)}{P(E)} $$

7. Final calculations for other parts

   Integrate the results to find:

   • $P(E \cap F)$, $P(G|E)$, etc., based on your defined outcomes.