<p>A supermarket sells 2 different types of apples – Granny Smith and Fuji. The masses, in grams, of each type of apples follow normal distributions. The means and standard deviations of these distributions are shown in the following table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 33px;"><th style="height: 33px; min-height: 33px; text-align: left; --cell-justify: flex-start"></th><th style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">Mean (g)</th><th style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">Standard deviation (g)</th></tr></thead><tbody><tr style="height: 33px;"><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">Granny Smith</td><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">150</td><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">11.7</td></tr><tr style="height: 33px;"><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">Fuji</td><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">180</td><td style="height: 33px; min-height: 33px; text-align: center; --cell-justify: center">15.2</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Two Fuji apples are chosen. Find the probability that their masses differs by at least 20 g.</div></li><li><div class="list-item-paragraph">Granny Smith apples are sold at \$3.50 per kg and Fuji apples at \$5 per kg.</div></li></ol>
<p>Find the probability that six Granny Smith apples and four Fuji apples cost more than $6.50.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">State an assumption needed for your calculations in parts <strong>(i)</strong> and <strong>(ii)</strong>.</div></li></ol>

A supermarket sells 2 different types of apples – Granny Smith and Fuji. The masses, in grams, of each type of apples follow normal distributions. The means and standard deviations of these distributions are shown in the following table.

Two Fuji apples are chosen. Find the probability that their masses differs by at least 20 g.

Granny Smith apples are sold at \$3.50 per kg and Fuji apples at \$5 per kg.

Find the probability that six Granny Smith apples and four Fuji apples cost more than $6.50.

State an assumption needed for your calculations in parts **(i)** and **(ii)**.

Normal Distribution - General

Normal Distribution

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

2015 VJC P2 Q11

Let G and F be the mass of a randomly chosen Granny Smith and Fuji apples respectively in grams.  $G \sim N(150, 11.7^2)$ $F \sim N(180, 15.2^2)$

$F_1 - F_2 \sim N(0, 2(15.2^2))$

Required Probability
$= P(|F_1 - F_2| \geq 20)$
$= 1 - P(|F_1 - F_2| < 20)$
$= 1 - P(-20 < F_1 - F_2 < 20)$
$= 0.3521$
$= 0.352$ (3 s.f.)

Total cost, $C = \frac{3.50}{1000}(G_1 + G_2 + \ldots + G_6) + \frac{5}{1000}(F_1 + \ldots + F_4)$

$E(C) = \frac{3.5}{1000}[6E(G)] + \frac{5}{1000}[4E(F)]$
$= \frac{3.5}{1000}(6)(150) + \frac{5}{1000}(4)(180)$
$= 6.75$

$\text{Var}(C) = \left(\frac{3.5}{1000}\right)^2 \text{Var}(G_1) + \ldots + \left(\frac{5}{1000}\right)^2 \text{Var}(F_1) + \ldots$

$= 6\left(\frac{3.5}{1000}\right)^2 \text{Var}(G) + 4\left(\frac{5}{1000}\right)^2 \text{Var}(F)$

$= 6\left(\frac{3.5}{1000}\right)^2 (11.7^2) + 4\left(\frac{5}{1000}\right)^2 (15.2^2)$

$= 0.033165$

$\therefore C \sim N(6.75, 0.033165)$

Required Probability $= P(C > 6.50)$
$= 0.91508$
$= 0.915$ (3 s.f.)

We assume that the masses of all the apples are independent of each other.

2015 VJC P2 Q11

2015 VJC P2 Q11

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