20N.2.SL.TZ0.T_2

pestleMathematicsAISLPaper 220N· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

On a school excursion, $100$ students visited an amusement park. The amusement park’s main attractions are rollercoasters ( $R$ ), water slides ( $W$ ), and virtual reality rides ( $V$ ).

The students were asked which main attractions they visited. The results are shown in the Venn diagram.

A total of $74$ students visited the rollercoasters or the water slides.

Find the value of $a$ .

[2]

a.i.

Find the value of $b$ .

[2]

a.ii.

Find the number of students who visited at least two types of main attraction.

[2]

Write down the value of $n (R \cap W)$ .

[1]

Find the probability that a randomly selected student visited the rollercoasters.

[2]

d.i.

Find the probability that a randomly selected student visited the virtual reality rides.

[1]

d.ii.

Hence determine whether the events in parts (d)(i) and (d)(ii) are independent. Justify your reasoning.

[2]

Markscheme / solution

$74 - (32 + 12 + 10 + 9 + 5)$ OR $74 - 68$ (M1)

Note: Award (M1) for setting up a correct expression.

$(a =) 6$ (A1)(G2)

[2 marks]

a.i.

$100 - (74 + 18)$ (M1)

$100 - 92$ (M1)

$100 - (32 + 9 + 5 + 12 + 10 + 18 + 6)$ (M1)

Note: Award (M1) for setting up a correct expression. Follow through from part (a)(i) but only for $a \geq 0$ .

$(b =) 8$ (A1)(ft)(G2)

Note: Follow through from part(a)(i). The value of $b$ must be greater or equal to zero for the (A1)(ft) to be awarded.

[2 marks]

a.ii.

$9 + 5 + 12 + 10$ (M1)

Note: Award (M1) for adding $9, 5, 12$ and $10$ .

$36$ (A1)(G2)

[2 marks]

$14$ (A1)

[1 mark]

$\frac{58}{100} (\frac{29}{50}, 0.58, 58 %)$ (A1)(A1)(G2)

Note: Award (A1) for correct numerator. Award(A1) for the correct denominator. Award (A0) for $58$ only.

[2 marks]

d.i.

$\frac{45}{100} (\frac{9}{20}, 0.45, 45 %)$ (A1)(ft)

Note: Follow through from their denominator from part (d)(i).

[1 mark]

d.ii.

they are not independent (A1)(ft)

$\frac{58}{100} \times \frac{45}{100} \neq \frac{17}{100}$ OR $0.261 \neq 0.17$ (R1)

Note: Comparison of numerical values must be seen for (R1) to be awarded.
Do not award (A1)(R0). Follow through from parts (d)(i) and (d)(ii).

[2 marks]

Examiners’ report

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a.i.

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a.ii.

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d.i.

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d.ii.

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