Probability distribution of discrete random variables (table + applications): notes and practice questions

In a security system, two independent sensors, System A and System B, report a threat level for an incident. System A reports a level $L_A \in \{1, 2, 3\}$ with probabilities $P(L_A=1)=0.2$, $P(L_A=2)=0.5$, $P(L_A=3)=0.3$. System B reports a level $L_B \in \{1, 2, 3, 4\}$ with probabilities $P(L_B=1)=0.1$, $P(L_B=2)=0.3$, $P(L_B=3)=0.4$, $P(L_B=4)=0.2$. The overall threat assessment, $T$, for an incident is defined as the higher of the two reported threat levels (i.e., $T = \max(L_A, L_B)$ ).

Complete the following table to show the probability distribution of $T$.

Find the probability that an incident has an overall threat assessment of at least 3.

Given that the overall threat assessment is at least 3, find the probability that System A reported a level of 2.

Calculate the expected overall threat assessment, $E(T)$.

(a) A company produces a new board game that includes a four-sided spinner. The spinner is designed to be fair, meaning each side (labelled 1, 2, 3, 4) should have an equal probability of being spun. During a quality control test, the spinner is spun $120$ times. The observed frequencies are:

Find the expected frequencies for each number if the spinner is fair.

(b) Write down the number of degrees of freedom for this test.

(c) The critical value for a goodness of fit test at the $1\%$ significance level with the appropriate degrees of freedom is $11.345$.

Determine the results of a goodness of fit test to find out whether the observed data fits a uniform distribution. Remember to write down the null and alternative hypotheses.

A manufacturing plant produces a specific electronic component. Due to the complexity of the manufacturing process, there is a 15% chance that any given component will be defective. A batch of 12 components is randomly selected for quality control inspection. Each component's defect status is independent of others.

(a) Calculate the expected number of components in the batch that are non-defective.

(b) Calculate the probability that exactly 3 components in the batch are defective.

(c) Determine the probability that more than 9 components in the batch are non-defective.

A game involves a player attempting to hit a target. The number of successful hits, $X$, in a round follows a discrete probability distribution given by:

$X$

$0$

$1$

$2$

$3$

$P(X=x)$

$\frac{1}{8}$

$\frac{3}{8}$

$\frac{3}{8}$

$\frac{1}{8}$

(a) Find $P(X \le 1)$.

(b) Giving your answers as fractions, calculate:

(i) $E(X)$

(c) The variance of $X$ is $\frac{3}{4}$. Let the random variable $Y = 5X$.

(i) Find $E(Y)$.

(ii) Find $Var(Y)$.

(d) Let the random variable $T = X_1 + X_2 + X_3 + X_4$, be the total of four independent values of $X$.

(i) Find $E(T)$.

(ii) Find $Var(T)$.

(e) Let the random variable $R = \frac{1}{X+1}$.

(i) Calculate $E(R)$ giving the answer as a fraction.

(ii) Hence, determine whether or not the statement $E\left(\frac{1}{X+1}\right) = \frac{1}{E(X)+1}$ is true. You must justify your answer.

A high-end watch manufacturer implements a quality control system. The average number of defective watches, $D$, in a standard production batch is related to the quality control score, $Q$, by the equation $\log_{10}D = k - Q$, where $k \in \mathbb{R}$.

For a quality control score of $Q = 2.5$, the manufacturer observes an average of $200$ defective watches per standard batch.

(a) Find the value of $k$.

(b) The equation relating $D$ and $Q$ can also be expressed in the form $D = \frac{m}{10^Q}$.

Find the value of $m$.

(c) The manufacturer aims for an extremely high quality control score of $Q = 7.8$.

Find the average number of defective watches in a standard batch for this quality score.

(d) The number of defective watches in a production run can be modelled by a Poisson distribution. The average number of defective watches in a standard batch (from part (c) ) represents the mean for a single production unit.

A large order consists of $250$ such standard production batches. Find the probability that this entire large order has no defective watches.

A factory produces specialized electronic components. The total "quality score" of a component, $T$, is a combination of scores from three independent inspection stages:

• Stage 1: Automated visual inspection. The score $X$ from this stage has an expectation of $2.5$ and a standard deviation of $0.7$.

• Stage 2: Manual functional test. A batch of $8$ critical functions are tested, and the score $Y$ is the number of functions that pass. Each function has a $0.35$ probability of passing, independently.

• Stage 3: Environmental stress test. The score $Z$ from this stage, representing the number of successful stress cycles, follows a Poisson distribution with a mean of $4.2$.

The overall quality score for a component is given by $T = X + 2Y + Z$.

Calculate the expected value and variance of the total quality score $T$.

Given that the distribution of $T$ can be approximated by a Normal distribution, find the probability that a randomly selected component has a total quality score between $10$ and $14$ (inclusive of $10$, exclusive of $14$).

The factory manager wants to ensure that the mean total quality score of a sample of $n$ components is within $0.8$ units of the true mean, with a probability of at least $0.95$. Find the minimum sample size $n$ required.

The number of incoming calls received by a tech support hotline in a particular 30-minute period is historically known to follow a Poisson distribution with a mean of 48 calls per hour.

A new automated system is implemented, and it is claimed that this system has decreased the number of calls requiring human intervention.

To test this claim, the number of calls, Y, requiring human intervention in a 30-minute period on a particular day will be recorded. The test will have the following hypotheses:

$H_0$: the mean number of calls requiring human intervention has not changed,

$H_1$: the mean number of calls requiring human intervention has decreased.

The alternative hypothesis will be accepted if $Y \leq 17$.

Assuming the null hypothesis to be true, state the distribution of Y.

Find the probability of a Type I error.

Find the probability of a Type II error, if the number of calls now follows a Poisson distribution with a mean of 35 calls per hour.

A baker attempts to bake a batch of 4 special occasion cakes. The random variable X represents the number of perfectly baked cakes in a batch. The probability distribution for X is given in the following table.

(a) Find the value of k.

The baker incurs costs and makes profit based on the number of perfectly baked cakes. The net gain (profit/loss) for the baker, in dollars, for a batch is shown in the following table.

(b) Determine whether, on average, the baker expects to make a profit or a loss from baking a batch of these cakes. Justify your answer.

Two unbiased dice, each with faces numbered from 1 to 6 inclusive, are rolled. The numbers on the uppermost faces of the dice are noted.

Let the random variable $X$ be the sum of the numbers on the dice.

(a)(i) Find $P(X=7)$.

(a)(ii) Find $P(X=10)$.

(b) Complete the table to show the probability distribution of $X$.

The probability that $X=5$ is shown.

(c) Calculate $E(X)$.

(d) Given that the sum of the numbers on the dice is a prime number, find the probability that $X=7$.

A player participates in a dart game where they throw one dart at a target with three distinct regions: an inner ring, a middle ring, and an outer ring. The probability of hitting each region in any given throw is shown in the following table.

The score awarded for hitting each region is:

• Inner Ring: 2 points

• Middle Ring: 1 point

• Outer Ring: 0 points

The player throws two darts independently. Find the probability that the player achieves a total score of exactly 2 points from the two throws.

In a different version of the game, the player wins $k$ points if they hit the Inner Ring, wins 4 points if they hit the Middle Ring, and loses 8 points if they hit the Outer Ring.

Find the value of $k$ such that the game is fair.

A board game involves drawing cards from a special deck. Each card has a score printed on it. The possible scores are $-5, -2, 0, 3, 4, 6$. The following table shows the probability distribution for the score, $X$, when a card is drawn.

Score $x$

$-5$

$-2$

$0$

$3$

$4$

$6$

$P(X=x)$

$\frac{1}{15}$

$p$

$\frac{2}{15}$

$\frac{3}{15}$

$\frac{4}{15}$

$\frac{1}{15}$

(a) Find the exact value of $p$.

(b) Calculate the expected score when drawing one card.

A player draws two cards from the deck, replacing the first card before drawing the second. Find the probability that the total score from the two cards is $1$.

A company manufactures microchips. On average, 1 in 20 microchips fails the final quality control test. The failure of any microchip is independent of others.

In a randomly selected batch of 30 microchips, calculate the probability that exactly three microchips fail quality control.

In a randomly selected batch of 30 microchips, calculate the probability that more than two microchips fail quality control.

The company sells functional microchips for $75 each and can sell failed microchips (for parts/rework) for$20 each.

Calculate the expected revenue, in dollars, from selling a randomly selected batch of 30 microchips.

The 'SoundWave' concert hall has a capacity of $100$ seats. Historical data shows that $90\%$ of ticket holders attend the concert. The management decides to sell $105$ tickets, hoping that no more than $100$ concertgoers will arrive.

The number of concertgoers who arrive is assumed to follow a binomial distribution with a probability of $0.9$.

(a) Calculate the probability that more than $100$ concertgoers arrive for the performance.

(b) (i) Write down the expected number of concertgoers who will arrive if $100$ tickets are sold.

(ii) Find the maximum number of tickets that could be sold if the expected number of concertgoers who arrive must be less than or equal to $100$.

Each ticket costs $120. If more concertgoers arrive than there are seats, the venue will give$200 in compensation to each concertgoer who cannot be seated.

(c) Find, to the nearest integer, the expected increase or decrease in the money made by the venue if they decide to sell $105$ tickets rather than $100$.

A game involves two stages. First, a player draws a marble from a bag containing 3 red marbles and 2 blue marbles. After drawing a marble, the player spins a fair three-sided spinner with sections labeled 1, 2, and 3.

If a red marble is drawn, the player's final score is the number shown on the spinner.

If a blue marble is drawn, the player's final score is two more than the number shown on the spinner.

Find the probability that a player's final score is 4.

Complete the following table, showing the probability distribution of the final score.

Calculate the expected value of the player's final score.

A customer service department receives contacts from two independent sources: phone calls and emails. The number of phone calls received per hour can be modelled by a Poisson distribution with a mean of $5.5$. The number of emails received per hour can be modelled by a Poisson distribution with a mean of $3.8$.

(a)(i) Find the probability that the department receives at most $20$ phone calls in a $3$-hour period.

(a)(ii) Find the probability that the department receives a total of more than $12$ contacts (phone calls and emails combined) in one hour.

Each phone call takes an average of $4$ minutes to handle, and each email takes an average of $7$ minutes to handle. Let $M$ be the total time (in minutes) spent handling contacts in one hour.

(b)(i) Find $E(M)$.

(b)(ii) Find $Var(M)$.

(c) State one reason why the distribution of $M$ cannot be a Poisson distribution.

(d) The customer service department operates for an $8$-hour shift. Use the Central Limit Theorem to find the probability that the mean time spent per hour handling contacts during this shift is greater than $45$ minutes.

A tech startup is launching a new app and offers a reward program for early users. Users can achieve different statuses based on their engagement.

Let $X$ be the net reward (in points) a user receives. The probability distribution for $X$ is shown below:

(a) Find the value of $p$.

(b) The startup expects to sign up $300$ new users. Find the expected number of users who will receive a penalty (score $-15$ points).

(c) The startup wants the reward program to be 'balanced', meaning the average net reward per user is $0$ points. Calculate the value of $w$ for this condition.

Tech Solutions Inc. is a customer support company. They are analyzing their call center efficiency. Over a long period, they collect data on the number of calls, $C$, already in the queue when a new customer's call arrives. The probability distribution of $C$ is shown in the following table.

Find the probability that there are at least two calls in the queue when a new customer's call arrives.

Find E($C$).

The time in seconds, $T$, taken to resolve a single customer's issue can be modelled by the normal distribution $T \sim \text{N}(130, 25^2)$.

The company's management estimates that the expected total time a new customer will wait before their issue is resolved can be found by calculating E($C$) $\times$ E($T$).

Find the value of E($C$) $\times$ E($T$).

The company considers a service time to be 'long' if it takes more than three minutes to resolve a single customer's issue.

Using the distribution of $T$ given above, find the probability that it takes more than three minutes to resolve a randomly selected customer's issue.

Find the probability it takes more than four minutes in total to resolve two randomly selected customers' issues. You may assume all service times are independent of all other service times.

The company assumes that when a new customer's call arrives, the person at the front of the queue has only just reached a support agent. They also assume that if there are three or more customers already in the queue, the new customer will definitely wait more than three minutes before being served.

Using these assumptions and the probabilities for $C$ given in the table above,

find the probability a customer just arriving at the call center will wait more than three minutes before being served.

The company has a policy to employ more staff if the probability that a customer has to wait more than three minutes before being served is greater than $0.35$.

Hence state whether Tech Solutions Inc. will decide to employ more staff.

(a) Find the value of $p$.

A fantasy adventure game involves exploring ancient ruins, where players can discover treasures or encounter perils. Points are awarded based on the outcome of each exploration:

• Discovering a rare relic scores $w$ points.

• Finding a common artifact scores $0$ points.

• Triggering a trap results in a loss of $8$ points.

Let $X$ be the number of points scored during a single exploration. The probability distribution is shown in the table below.

(b) The ruins are explored $80$ times. Find the expected number of times a player triggers a trap.

(c) Calculate the value of $w$, given that the game is fair.