Two independent events F and S are represented in the Venn diagram shown below - Leaving Cert Mathematics - Question 6 - 2019

Given that the events F and S are independent:

1. Sum of probabilities: We use the formula:

   $P(F \cup S) = P(F) + P(S) - P(F \cap S)$

   From the question, we have:

   $P(F \cap S) = \frac{1}{4}, \ P(F \cap S') = \frac{1}{5}$

   Let:

   • $P(F) = a$
   • $P(S) = b$.

   This implies:

   $P(F \cap S') = a - P(F \cap S) = a - \frac{1}{4}$

   Hence, for event S, we have: $P(S \cap F') = b - P(F \cap S) = b - \frac{1}{4}$

   Since the events are independent:
   $P(F \cap S') = P(F) imes P(S') => \frac{1}{5} = a(1 - b)$

   Next, we can derive: $P(F \cap S) = P(F) \times P(S) = a \times b = \frac{1}{4}.$

   From here, substituting values will give us $x$ and $y$.

2. Finding values: To find $x$ and $y$ more specifically, we can set up the equations based on the information:

   $\frac{1}{4} = ab \quad (1)$ $\frac{1}{5} = a(1 - b) \quad (2)$

   Replacing $a$ from (1) into (2) gives value of $x = \frac{9}{20}$ and $y = \frac{11}{45}$.