4.7 Discrete random variables questions for IB Maths AI

Find
(i) $\mathrm{E}(X)$;
(ii) $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Zoltan takes 40 turns. Use the central limit theorem to find the probability that he moves more than 120 squares in total.

Write down the value of $p$.

Find
(i) $\mathrm{E}(X)$;
(ii) $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Let $G$ be the charity's profit, in NZD, from one game, so that $G=6-X$. Find the mean and the standard deviation of $G$.

The game is played 200 times, and the games are independent.

(i) Find the mean and the standard deviation of the charity's total profit.
(ii) Use the central limit theorem to estimate the probability that the charity makes a loss.

The organiser, Tama, suggests doubling every prize and charging 12 NZD per game, so that the profit from one game is $H=12-2X$.

(i) Find $\mathrm{E}(H)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(H)$.
(ii) Tama claims that this change will double the expected profit from 200 games without increasing the risk of a loss. Comment on his claim.

Find the probability that a randomly chosen tile is rejected.

Before inspection, tiles are packed in boxes of 25. The thicknesses of tiles are independent.

Find the expected number of tiles in a box that would be rejected.

Find the probability that a box contains at most one tile that would be rejected.

The manager, Farida, wants no more than 1 % of tiles to be rejected. The mean thickness stays at 9.2 mm.

Find the largest standard deviation of the thickness that achieves this.

The masses of the tiles are independent and normally distributed with mean 410 g and standard deviation 12 g. The mass of an empty box is normally distributed with mean 350 g and standard deviation 20 g.

Find the probability that a full box of 25 tiles has a total mass of more than 10.7 kg.

Every hour, an inspector measures the thickness of a random sample of 16 tiles and calculates the sample mean, $\bar{T}$. The machine is stopped if $\bar{T}$ lies outside the interval $9.2-k<\bar{T}<9.2+k$, where $k$ is chosen so that, when the mean thickness is 9.2 mm, $\mathrm{P}(9.2-k<\bar{T}<9.2+k)=0.99$.

Find the value of $k$.

This procedure is a hypothesis test about $\mu$, the mean thickness of the tiles being made.

State the null and alternative hypotheses and the significance level of the test.

One hour, the sample mean is 9.27 mm.

State, with a reason, whether the machine should be stopped.

Write down the probability that the machine is stopped when the mean thickness is in fact 9.2 mm.

A fault causes the mean thickness to drift to 9.3 mm, while the standard deviation stays at 0.15 mm.

Find the probability that the next hourly sample does not lead to the machine being stopped.

Find the proportion of tiles that would be rejected when the mean thickness is 9.3 mm.

Hence comment on the effectiveness of the hourly check.

Find the value of $k$.

Find the probability that the team scores at least two goals in a match.

Find $\mathrm{E}(X)$.

Find the value of $k$.

Find $\mathrm{E}(W)$.

The profit for the charity from one game is $Y=5-W$. Show that $\mathrm{E}(Y)=1.05$ EUR.

Find $\mathrm{V}\mathrm{a}\mathrm{r}(W)$ and hence write down $\mathrm{V}\mathrm{a}\mathrm{r}(Y)$.

During the fair the game is played 200 times. Let $T$ be the total profit for the charity, in EUR.

Use the central limit theorem to write down the approximate distribution of $T$, giving reasons why the theorem applies.

Find the probability that
(i) the total profit is more than 250 EUR;
(ii) the charity makes a loss.

Tomasz says that the game is a safe way to raise money. Comment on his statement.

Show that $k=\frac{1}{18}$.

Find $\mathrm{E}(X)$. Give your answer as a fraction.

Given that at least two chicks hatch in a nest, find the probability that exactly four chicks hatch.