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

Given that B and D are independent events, we can express the relationship as:

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

The Venn diagram tells us that:

• $P(B) = 0.6$,
• $P(D) = 0.27 + 0.15 + t$.

By using independence:

$P(B) imes P(D) = P(B) imes (0.27 + 0.15 + t)$ Substituting in gives:

$0.6 imes (0.42) = 0.27 + 0.15 + t$ So, $0.6 imes 0.42 = 0.27 + 0.15 + t$ This results in:

$t = 0.018$

Using the probabilities already found, we can determine $u$. From the earlier calculation we have:

$u = 1 - (0.6 + 0.15 + t)$ Substituting $t$ gives us:

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

Using the definition of conditional probability:

$P(D ext{ and } R | B) = \frac{P(D ext{ and } R)}{P(B)}$ From the diagram we know:

• $P(D ext{ and } R) = 0.27$ Thus:

$P(D ext{ and } R | B) = \frac{0.27}{0.6} = 0.45$

In this case:

$P(D | R ext{ and } B') = \frac{P(D ext{ and } R ext{ and } B')}{P(R ext{ and } B')}$ Using values from the Venn diagram:

• $P(D ext{ and } R ext{ and } B') = 0.15$
• $P(R ext{ and } B') = 0.15 + 0.33 = 0.48$ Finally:

$P(D | R ext{ and } B') = \frac{0.15}{0.48} ext{ which evaluates approximately to } 0.3125$

From the total of 77 customers:

• 40 have booked a room and breakfast
• 37 have booked a room without breakfast Thus, the expected number of customers booking dinner can be approximated by:

egin{align*} ext{Total customers} & = 77\ ext{Booked dinner} & = 0.75 imes 77 ext{ (estimated proportion for dinner)} ext{ which equals } 33 ext{ customers.} ext{Thus, estimated are } 33 ext{ customers.} ext{Conclusion: } 33 customers are estimated to book dinner.