<p>A student reaches the bus-stop outside his house at $7.20$ am every morning to take a bus to school and has to reach his school by $7.40$ am. Assume that the waiting time for his bus is normally distributed with mean $8$ minutes and variance $5$ minutes$^{2}$, and the journey time is normally distributed with mean $11$ minutes and variance $4$ minutes$^{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that he will take more than $20$ minutes to reach school (i.e. late for school) on a randomly chosen day.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">In a month of $30$ days, what is the expected number of days he will be late for school?</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the latest time he would have to reach the bus-stop outside his house so that the probability of him being late for school is less than $5$%.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the probability that the average time taken to travel from the bus stop outside his house (including waiting for the bus) to school in $30$ days is between $15$ to $20$ minutes.</div><p style="text-align: right;">[2]</p></li></ol>

A student reaches the bus-stop outside his house at $7.20$ am every morning to take a bus to school and has to reach his school by $7.40$ am. Assume that the waiting time for his bus is normally distributed with mean $8$ minutes and variance $5$ minutes$^{2}$, and the journey time is normally distributed with mean $11$ minutes and variance $4$ minutes$^{2}$.

Find the probability that he will take more than $20$ minutes to reach school (i.e. late for school) on a randomly chosen day.

In a month of $30$ days, what is the expected number of days he will be late for school?

Find the latest time he would have to reach the bus-stop outside his house so that the probability of him being late for school is less than $5$%.

Find the probability that the average time taken to travel from the bus stop outside his house (including waiting for the bus) to school in $30$ days is between $15$ to $20$ minutes.

Normal Distribution - General

Normal Distribution

International Baccalaureate Higher Level Mathematics

IB HL Math

CJC P2 Q9

Let $J$ and $W$ be the journey time and waiting time, in minutes, respectively.

$J \sim \text{N}(11, 4)$, $W \sim \text{N}(8, 5)$

Required Probability

$= \text{P}(W + J > 20)$

$W + J \sim \text{N}(19, 9)$

$= 0.36944$

$= 0.369$ (3 s.f.)

$\text{E}(\text{late}) = np = 30(0.36944) = 11.08 = 11$ days

Let $a$ be the travel time.

$\text{P}(W + J > a) < 0.05$

$a > 23.93$ min

$\therefore$ Latest time to leave $= 7{:}40\text{ am} - 23.93\text{ minutes} = 7{:}16\text{ am}$

Let $X = W + J$.

$n = 30$, $\bar{X} \sim \text{N}\!\left(19,\; \dfrac{9}{30}\right)$

Required Probability $= \text{P}(15 < \bar{X} < 20) = 0.966$ (3 s.f.)

CJC P2 Q9

CJC P2 Q9

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