Let's analyze and solve the problem step by step:

Problem a)

You are asked to prove the following formula: $P(A \cap B) = P(A) - P(A \setminus B) + P(A \mid B) \cdot P(B)$

Key concepts to use:

1. $P(A \cap B) = P(A) - P(A \setminus B)$: This comes from the fact that $A = (A \setminus B) \cup (A \cap B)$, and $(A \setminus B)$ and $(A \cap B)$ are disjoint sets.

2. $P(A \cap B) = P(A \mid B) \cdot P(B)$: This is the definition of conditional probability.

We combine these two ideas to prove the given equality. Here's the outline:

• Decompose $P(A)$ into its contributions from $A \setminus B$ and $A \cap B$.
• Use the definition of conditional probability.

Would you like me to provide a fully written mathematical proof of this?

Problem b1)

From the problem:

• $\frac{1}{4}$ of the girls have green eyes.
• $\frac{1}{3}$ of the girls have blonde hair.
• Of the blonde-haired girls, $\frac{1}{2}$ have green eyes.

You are tasked to find the probability that a randomly chosen girl is not blonde and does not have green eyes.

Approach:

1. Use the information to calculate $P(\text{blonde})$, $P(\text{green eyes})$, and $P(\text{blonde} \cap \text{green eyes})$.
2. Use complementary probabilities for $P(\text{not blonde})$ and $P(\text{not green eyes})$.
3. Subtract overlaps to find $P(\text{not blonde} \cap \text{not green eyes})$.

Problem b2)

There are 120 girls in Vale do Rei, and you need to form a committee of 5 girls such that exactly two have blonde hair.

Approach:

1. Calculate the number of blonde-haired girls ($\frac{1}{3} \cdot 120 = 40$).
2. Calculate the number of non-blonde-haired girls ($120 - 40 = 80$).
3. Use combinations to determine:
   • Ways to select 2 blonde-haired girls: $\binom{40}{2}$.
   • Ways to select 3 non-blonde-haired girls: $\binom{80}{3}$.
4. Multiply the results to find the total number of valid committees.

Would you like me to provide detailed calculations and solutions for each part?