19M.2.AHL.TZ1.H_9

pestleMathematicsAAHLPaper 219M· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

A café serves sandwiches and cakes. Each customer will choose one of the following three options; buy only a sandwich, buy only a cake or buy both a sandwich and a cake.

The probability that a customer buys a sandwich is 0.72 and the probability that a customer buys a cake is 0.45.

Find the probability that a customer chosen at random will buy

On a typical day 200 customers come to the café.

It is known that 46 % of the customers who come to the café are male, and that 80 % of these buy a sandwich.

both a sandwich and a cake.

[3]

a.i.

only a sandwich.

[1]

a.ii.

Find the expected number of cakes sold on a typical day.

[1]

b.i.

Find the probability that more than 100 cakes will be sold on a typical day.

[3]

b.ii.

A customer is selected at random. Find the probability that the customer is male and buys a sandwich.

[1]

c.i.

A female customer is selected at random. Find the probability that she buys a sandwich.

[4]

c.ii.

Markscheme / solution

use of formula or Venn diagram (M1)

0.72 + 0.45 − 1 (A1)

= 0.17 A1

[3 marks]

a.i.

0.72 − 0.17 = 0.55 A1

[1 mark]

a.ii.

200 × 0.45 = 90 A1

[1 mark]

b.i.

let X be the number of customers who order cake

X ~ B(200,0.45) (M1)

P(X > 100) = P(X ≥ 101)(= 1 − P(X ≤ 100)) (M1)

= 0.0681 A1

[3 marks]

b.ii.

0.46 × 0.8 = 0.368 A1

[1 mark]

c.i.

METHOD 1

$0.368 + 0.54 \times {\text{P}}\left( {S\left| F \right.} \right) = 0.72$ M1A1A1

Note: Award M1 for an appropriate tree diagram. Award M1 for LHS, M1 for RHS.

${\text{P}}\left( {S\left| F \right.} \right) = 0.652$ A1

METHOD 2

${\text{P}}\left( {S\left| F \right.} \right) = \frac{{{\text{P}}\left( {S \cap F} \right)}}{{{\text{P}}\left( F \right)}}$ (M1)

$= \frac{{0.72 - 0.368}}{{0.54}}$ A1A1

Note: Award A1 for numerator, A1 for denominator.

${\text{P}}\left( {S\left| F \right.} \right) = 0.652$ A1

[4 marks]

c.ii.

Examiners’ report

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a.i.

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a.ii.

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b.i.

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b.ii.

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c.i.

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c.ii.