The Venn diagram shows the probabilities of customer bookings at Harry’s hotel - Edexcel - A-Level Maths Statistics - Question 4 - 2016 - Paper 1

Since events B and D are independent, we can use the relation:

$P(B ext{ and } D) = P(B) imes P(D)$

From the Venn diagram:

• Total probability of B = 0.6
• Total probability from the diagram = 0.27 + 0.15 + t = 0.42.

Using independence, we have:

$0.27 + 0.15 + t = 0.27 imes D$

This results in $0.6 imes (0.42 + t) = 0.27$. Solving for t gives: $t = (0.27 / 0.6) - 0.42 = 0.018$.

Using the probabilities derived, we have:

$u = 1 - (0.6 + 0.15 + t)$ Substituting t = 0.018,

$u = 1 - (0.6 + 0.15 + 0.018) = 0.22$.

To calculate this probability, we use the formula:

$P(D | R ∩ B) = \frac{P(D ∩ R ∩ B)}{P(R ∩ B)}$ From the diagram:

• Probability of D ∩ R ∩ B = 0.27,
• Probability of R ∩ B = 0.42. Thus:

$P(D | R ∩ B) = \frac{0.27}{0.42} = 0.643$.

Again using the conditional probability formula:

$P(D | R ∩ B') = \frac{P(D ∩ R ∩ B')}{P(R ∩ B')}$ From the diagram:

• Probability of D ∩ R ∩ B' = 0.15,
• Probability of R ∩ B' = 0.15 + 0.033 = 0.15. Thus:

$P(D | R ∩ B') = \frac{0.15}{0.15} = 1.0$.

From part (a), we found that the total probability of booking dinner is:

Using total customers, we find: $dinner ext{ bookings} ext{ } = 77 imes 0.45 = 33$. Thus, approximately 33 customers are expected to book dinner.