Probability distribution tables for discrete random variables: notes and practice questions

On a Saturday at a cinema, a sample of 50 customers was randomly selected. They were asked how many snack items they had purchased. This information is summarized in the following frequency table.

It can be assumed that this sample is representative of all customers for the following day.

For the following day, Sunday, estimate

(i) the probability that a randomly selected customer will purchase at least one snack item.

For the following day, Sunday, estimate

(ii) the expected number of snacks purchased by a customer.

It is known that 800 customers will attend the cinema on Sunday. The average price of a snack item is $5.50.

Estimate the total revenue from snack sales on Sunday.

A spinner with four sectors is spun. The sectors are numbered 1, 2, 3, and 4. Let $X$ be the score obtained when the spinner is spun. The probability distribution for $X$ is given in the following table.

(a) Find the value of $k$.

(b) Find the value of $E(X)$.

A second spinner, B, is also spun. Let $Y$ be the score obtained. The probability distribution for $Y$ is given in the following table.

(c) (i) State the range of possible values of $n$.

(ii) Hence, find the range of possible values of $m$.

(d) Hence, find the range of possible values for $E(Y)$.

Leo spins spinner A once and Mia spins spinner B once. The probability that Leo's score is greater than Mia's score is $\frac{2}{5}$.

(e) Find the value of $E(Y)$.

A local coffee shop, "The Daily Grind", surveyed a random sample of 50 customers about their purchasing habits over one week. The following table shows the number of coffees purchased by these customers.

This sample is considered representative of all customers for the following week.

For the following week, estimate:

(a.i) the probability that a randomly selected customer will purchase at least one coffee;

(a.ii) the expected number of coffees a customer will purchase.

(b) The coffee shop expects to serve 800 customers in the following week. Each large batch of coffee brewed can serve a maximum of 20 cups.

Estimate the minimum number of large batches of coffee that must be brewed to meet the expected demand.

A biased four-sided spinner, A, is spun. Let $S_A$ be the score obtained. The probability distribution for $S_A$ is given in the following table.

Find the value of $k$.

Hence, find the value of $E(S_A)$.

A second biased four-sided spinner, B, is spun. Let $S_B$ be the score obtained. The probability distribution for $S_B$ is given in the following table.

(i) State the range of possible values of $b$.

(ii) Hence, find the range of possible values of $a$.

Hence, find the range of possible values for $E(S_B)$.

Leo spins spinner A once and Mia spins spinner B once. The probability that Leo's score is greater than Mia's score is $\frac{13}{49}$.

Find the value of $E(S_B)$.

Consider a geometric sequence with $u_1 = 16$ and $u_4 = 2$.

(a) Find the common ratio, $r$.

(b) The following table shows the probability distribution of a discrete random variable $Y$ such that $P(Y=n) = \frac{u_n}{k}$, where $n \in \{1, 2, 3, 4\}$ and $k \in \mathbb{R}^+$.

Find the value of $k$.

All answers in this question should be given to four significant figures.

A popular online game offers players a 'Mystery Box' for £5. Each box contains a prize, with the probability distribution for the prize value $D$ shown in the following table. For example, the probability of a player receiving £$25$ is 0.04. The initial grand prize in the first week of the game is £$500$.

(a) Find the value of $c$.

(b) Determine whether purchasing a mystery box in the first week is a fair game. Justify your answer.

(c) If the grand prize is not won and continues to triple each week, while all other prize amounts and probabilities remain the same, write an expression in terms of $n$ for the value of the grand prize in the $n$th week of the game.

(d) The $w$th week is the first week in which a player is expected to make a profit from purchasing a mystery box. If a player purchases a mystery box in the $w$th week, their expected profit is $p$.

Find the value of $p$.

A spinner has four sectors. When the spinner is spun, the score, Y, is a discrete random variable. The probability distribution of Y is given in the following table:

(a) Find the range of possible values of k.

(b) In the case where $k = 0.1$, determine $\operatorname{Var}(3Y+1)$.

(a) The random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Find $P(\mu - 1.2\sigma < X < \mu + 1.2\sigma)$.

(b) The diameters of ball bearings produced by a factory, in mm, are normally distributed with mean $\mu$ and standard deviation $\sigma$. The ball bearings are categorized as defective, standard, large, or premium, according to their diameter. The following table shows the probability a ball bearing is classified into each category.

The maximum diameter of a defective ball bearing is 14.8 mm.

The minimum diameter of a premium ball bearing is 16.5 mm.

Find the value of $\mu$ and of $\sigma$.

(c) The factory rejects all defective ball bearings. The remaining ball bearings are sold.

Find the probability that a ball bearing chosen at random from those sold is categorized as

(i) standard;

(ii) large;

(iii) premium.

(d) The selling prices of the different categories of ball bearings at this factory are shown in the following table:

The factory incurs a fixed cost of $300 for the production run and assumes it will sell the accepted ball bearings in exactly the same proportion as calculated in part (c).

According to this model, find the minimum number of accepted ball bearings that must be sold so that the net profit for the factory is at least $550.

A discrete random variable, X, represents the number of successful attempts in a particular game. The probability distribution for X is given in the table below:

Show that $k^2 - k + 0.16 = 0$.

Find the value of k, giving a reason for your answer.

Hence, find E(X).

(a) A batch of $N$ electronic components contains $D$ defective components and $F$ functional components. Components are selected randomly, one by one, without replacement.

(i) Find, in terms of $N$, the probability that the first component selected is defective.

(ii) Given that $N = 25$, $D = 10$, and $F = 15$, find the probability that the first two components selected are defective.

(b) Show that the probability that the first two components selected are functional is 0.35.

(c) Find the probability that the first three components selected are all functional.

(d) Find the probability that at least one of the first three components selected is defective.

(e) A technician earns 10 points if the first defective component is found on the third test, and 50 points if the first defective component is found on the fourth test. The technician tests $k$ such batches. Find the least value of $k$ such that the technician's expected total score is greater than 100.

A scientific team is studying the distribution of a rare particle in a specialized chamber. The probability density function for the distance $x$ (in cm) of a particle from the chamber's center is given by

$f(x) = \begin{cases} \frac{x}{\sqrt{(x^2+k)^3}} & 0 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases}$

where $k \in \mathbb{R}^+$.

Show that $\sqrt{25+k}-\sqrt{k} = \sqrt{k}\sqrt{25+k}$.

Find the value of $k$.

(a) The lifespan of a new type of battery, $L$ hours, can be modelled by a normal distribution with a mean of 1200 hours and a standard deviation of $\sigma$ hours.

Given that 5% of the batteries last longer than 1280 hours, find the value of $\sigma$.

(b) Find the probability that a randomly selected battery will have a lifespan of less than 1150 hours.

(c) Given that a battery lasts longer than 1150 hours, find the probability that it lasts less than 1250 hours.

(d) A batch of 500 batteries is produced. Find the expected number of batteries that will have a lifespan of less than 1150 hours.

(e) Find the probability that more than 85 of the batteries in this batch will have a lifespan of less than 1150 hours.

The lifespan, in years, of a new type of rechargeable battery is modelled by a continuous random variable $T$ with probability density function $f$ defined by

$f(t)=\begin{cases} \frac{3}{32}(-t^2 + 6t - 5), & 1 \leq t \leq 5 \\ 0, & \text{otherwise} \end{cases}$

Find the mode of $T$.

Find $\text{P}(1.5 \leq T \leq 3.5)$.

Find the median of $T$.

A manufacturer offers a warranty for these batteries. If a battery fails within the first $2$ years, it is replaced free of charge. However, if a battery lasts longer than $4$ years, it is considered a premium product and the customer receives a discount on their next purchase.

Find the probability that a randomly selected battery is either replaced under warranty or qualifies for a premium discount.

The manufacturer sells each battery for $120. If a battery fails before$2.5$years, the manufacturer incurs an additional warranty cost of$30. If it lasts $2.5$ years or longer, there is no additional cost.

Find the expected cost to the manufacturer per battery. Give your answer correct to the nearest cent.

Leo has a box of tokens which are coloured either silver, gold or bronze.

The box contains exactly 15 bronze tokens. The number of silver tokens is four times the number of gold tokens.

Leo plays a game where he takes 12 tokens out of the box, one at a time. He notes the colour of each token and returns it to the box before taking the next token out of the box.

The probability that the first token is bronze is 0.3.

Show that there are 28 silver tokens in the box.

Find the probability that at least 7 of the 12 tokens that Leo takes are bronze. Give your answer correct to five significant figures.

Leo has to pay $\$10$ to take part in the game. If he takes at least 7 tokens of the same colour, he wins a prize. If he does not take at least 7 tokens of the same colour, then he does not win a prize. There is a different prize for each colour, as shown in the following table, where $C \in \mathbb{Z}^+$.

Let the random variable $X$ represent Leo's net gain in dollars when he plays the game once. For example, if he takes at least 7 gold tokens, his net gain is $100 - 10 = 90$ since he gains $\$90$.

The probability distribution of $X$ is shown in the following table, with probabilities given correct to four decimal places, where $A \in \mathbb{Z}, q \in \mathbb{R}$.

Write down the value of $A$.

Use the probabilities in the table to find the value of $q$.

Determine the smallest integer value of $C$ for which Leo could expect to make a positive net gain.

Leo wants to play the game until he wins a prize.

Find the minimum number of times Leo needs to play the game in order that the probability of winning at least one prize is greater than 0.999.

A computer system has a list of $n$ possible passwords, three of which are classified as 'weak'. A hacker tries passwords from the list at random, one by one, without replacement. The system locks down when a weak password is tried.

Find the probability, in terms of $n$, that the system locks down on the first attempt.

Find the probability, in terms of $n$, that the system locks down on the second attempt.

Let $n=6$. Find the probability that the system will lock down on the

third attempt.

fourth attempt.

The hacker is part of a simulation where $n=6$. The simulation costs $24 to run. The hacker receives a payout depending on the number of attempts it takes to find a weak password. Let $P$ be the payout amount. This information is shown in the following table.

Find the value of $k$ so that this is a fair game.

A game development studio is analyzing the success rate of its new feature launches. Let Y be a discrete random variable representing the number of successful features launched in a quarter, with the following probability distribution:

Show that $2k^2 - k + 0.12 = 0$.

Find the value of $k$, giving a reason for your answer.

Hence, find E(Y).