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

Since B and D are independent events, we have:

$P(B)P(D) = P(B \cap D)$

From the diagram, we know:

$P(B) = 0.6, \, P(D) = 0.27, \, P(B \cap D) = 0.27 + 0.15 + t = 0.42$

Substituting:

$0.6 \times 0.27 = 0.42 \implies t = 0.018$

Now we can find the value of u. From the total probability condition, we have:

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

Calculating gives:

$u = 1 - 0.768 = 0.22$

Using the definition of conditional probability:

$P(D|R \cap B) = \frac{P(D \cap R \cap B)}{P(R \cap B)}$

From the Venn diagram, this is:

$P(D|R \cap B) = \frac{0.27}{0.27 + 0.33} = 0.45$

Again using the definition of conditional probability:

$P(D|R \cap B') = \frac{P(D \cap R \cap B')}{P(R \cap B')}$

From the values:

$P(D|R \cap B') = \frac{0.15}{0.15 + 0.33} = 0.45$

From the data provided, we need to find the proportion of customers who would book dinner. We know that:

40 customers booked a room and breakfast, and 37 booked a room only. This totals to:

$ext{Total customers booked breakfast (for dinner)} = 40 + 37 = 77$

Out of these, using the independent probability:

The expected number of customers who book dinner is given by:

$0.6 \times 77 = 33$