Probability basics (expected #, complementary events, probability of event): 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, $E(T)$.

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 company conducted performance reviews for 120 employees across two departments: Marketing and Engineering. The outcomes are summarized in the following table.

An employee is chosen at random from this group.

(a) Find the probability that the randomly chosen employee received an 'Excellent' rating.

(b) Given that the chosen employee received an 'Excellent' rating, find the probability that they work in the Marketing department.

(c) Two different employees are chosen at random from the original group. Find the probability that both employees work in the Marketing department.

The diameter of a certain type of industrial component is modelled by a normal distribution with a mean of $50$ mm and a standard deviation of $0.5$ mm.

Find the probability that a randomly selected component has a diameter less than $50.7$ mm.

Find the probability that a randomly selected component has a diameter greater than $51.2$ mm.

Assuming the diameters of components are independent, find the probability that two consecutive components both have a diameter greater than $51.2$ mm.

A component is classified as 'premium' if its diameter is between $49.5$ mm and $50.5$ mm. A batch of three components is considered 'high quality' if all three components in the batch are premium. Find the probability that a randomly selected batch is NOT high quality.

In a production run, $12$ batches of three components are produced. Find the probability that at least $7$ of these batches are high quality.

Find the probability that between $7$ and $10$ (exclusive of $10$) of these batches are high quality.

Given that at least $7$ batches are high quality, find the probability that less than $10$ batches are high quality.

A bakery produced a batch of 200 loaves of bread, consisting of sourdough and rye. The sales outcomes for these loaves are shown in the following table.

(a) Find the probability that a randomly chosen loaf from this batch was sold by the bakery.

A loaf is chosen at random from this batch. It is found that this loaf was sold.

(b) Find the probability that the loaf was a sourdough loaf.

Two different loaves are chosen at random from the original batch of 200 loaves.

(c) Find the probability that both loaves were rye.

(a) The number of customer service emails received by 'TechSolutions' per hour follows a Poisson distribution with a mean of $2.8$. Using this model, find the probability that TechSolutions receives exactly $3$ emails in a given hour.

(b) Over a period of $4$ consecutive hours, find the probability that TechSolutions receives:

(i) exactly $10$ emails.

(ii) emails during the first and third hour only (i.e., at least one email in the first hour, no emails in the second hour, and at least one email in the third hour).

(c) Over a working day of $10$ hours, find the probability that there are exactly $2$ hours during which TechSolutions receives no emails.

(d) TechSolutions expands its operations and opens a new department, 'SupportPlus', which also receives emails. The number of emails received by each 'SupportPlus' agent per hour follows a Poisson distribution with a mean of $1.5$. Assuming these are independent of the main TechSolutions department and each other, determine the least number of SupportPlus agents required so that the total probability of receiving at least $25$ emails across all departments (main TechSolutions and the SupportPlus agents) in an hour is greater than $0.15$.

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.

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) The manager of "The Daily Grind" coffee shop suggested that the number of customers arriving at the shop during a 5-minute interval can be modelled by a Poisson distribution.

Suggest two observations that the manager may have made that led him to suggest this model.

(b) Now assume that the model is valid and that the mean number of customers arriving at the shop during a 5-minute interval is $1.5$.

The manager observes customer arrivals during a 15-minute interval.

Calculate the probability that exactly 6 customers arrive during this 15-minute interval.

(c) Using the same model as in part (b), find the probability that fewer than 4 customers arrive during a 15-minute interval.

(d) Find the probability that in four consecutive 5-minute intervals, at least one customer arrives in each interval.

(e) Following a new marketing campaign, the manager wished to determine whether the mean number of customers arriving during a 5-minute interval had increased.

State the hypotheses for the test.

(f) Find the critical region for the test at the 5% significance level.

(g) Given that the mean number of customers per 5-minute interval has actually risen to $2.5$, find the probability that the manager makes a Type II error.

In a survey of students at a local college, it was found that the probability a student studies Art (event $A$) is $P(A) = 0.4$. The probability a student studies Biology (event $B$) is $P(B) = 0.7$. The probability that a student studies either Art or Biology or both is $P(A \cup B) = 0.8$.

(a) Calculate the probability that a randomly selected student studies Biology but not Art.

(b) Determine the probability that a randomly selected student studies Art or does not study Biology.

(c) Find the probability that a randomly selected student does not study both Art and Biology.

(a) A single data packet can be transferred between three servers, S1, S2, and S3, in a network. The possible direct transfers are:

• From S1, the packet can be sent to S2 or stay in S1.

• From S2, the packet can be sent to S1 or S3.

• From S3, the packet can be sent to S2 or stay in S3.

Write down the adjacency matrix for the directed graph representing these possible direct data transfers, ordering the servers S1, S2, S3.

(b) Find the total number of distinct data paths of length 5 from server S1 to server S2.

(c.i) Every possible sequence of 5 data transfers has the same probability of occurring. State this probability.

(c.ii) Use your answer to part (b) to find the probability that if a data packet was initially on server S1, it will be on server S2 after 5 transfers.

(d.i) A network administrator monitors the movement of two data packets. The possible combined states of the two packets (assuming they occupy distinct servers) are $S_{12}$ (packets on S1 and S2), $S_{13}$ (packets on S1 and S3), and $S_{23}$ (packets on S2 and S3).

The transitions between these states are modelled by the following transition matrix $T$, where rows represent the current state and columns represent the next state, in the order $S_{12}, S_{13}, S_{23}$:

$T = \begin{pmatrix} 0 & 0.25 & 0.4 \\ 0.6 & 0.5 & 0.6 \\ 0.4 & 0.25 & 0 \end{pmatrix}$

State the probability that if the packets are currently in state $S_{12}$, they will be in state $S_{13}$ after one transfer.

(d.ii) Using the transition matrix $T$ from part (d.i), state the probability that if the packets are currently in state $S_{13}$, they will be in state $S_{23}$ after one transfer.

(d.iii) Using the transition matrix $T$ from part (d.i), state the probability that if the packets are currently in state $S_{23}$, they will be in state $S_{12}$ after one transfer.

(e) Given that the two data packets are initially in state $S_{12}$ (on servers S1 and S2), find the probability that they will be in state $S_{23}$ (on servers S2 and S3) after 5 transfers.

(f) The data packets continue this pattern of transfers for a long period. Find the server that is occupied least and the proportion of the time it is free.

In a survey of $100$ students at a high school, data was collected on their subject choices. It was found that $65$ students were studying Mathematics, and $45$ students were studying Physics.

Every student surveyed was studying at least one of these two subjects.

(a) Determine how many students were studying both Mathematics and Physics.

(b) Find the probability that a randomly selected student studies Mathematics, but not Physics.

(c) Explain why the events "studying Mathematics" and "studying Physics" are not independent events.

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 group of students were surveyed about their participation in three extracurricular clubs: Environmental Club (E), Music Club (M), and Theater Club (T). The probabilities of their participation are represented in the Venn diagram below.

Justify that events M and T are not independent.

Explain why events E and T are mutually exclusive.

Determine whether events E and M are independent.

Determine whether events E' and M' are mutually exclusive.

Find P(T $\cap$ E').

A quality control team at a manufacturing plant inspects newly produced widgets. The probability that a single widget has a detectable defect is $0.15$. The inspections are independent events.

(a) Find an expression for the probability that at least one defect is detected in $n$ inspections.

(b) Hence, determine the least number of inspections required for the probability of detecting at least one defect to be greater than $99.9\%$.

A set of numbered cards is created, where each card has a number that is a multiple of $4$. The numbers start from $4$ and go up to $6n$, where $n$ is an even positive integer.

Calculate the total number of cards in the set, in terms of $n$.

Find the probability, in terms of $n$, that a card selected at random from the set shows a number that is divisible by $6$.

A collector has a box containing 20 vintage stamps. Of these, 4 are considered rare editions.

Three stamps are randomly selected from the box.

(a) Find the probability that exactly one of the selected stamps is a rare edition.

(b) Find the probability that at least one of the selected stamps is a rare edition.

Clara is taking three independent online quizzes: Mathematics, Physics, and Chemistry. The probability that she passes the Mathematics quiz is $0.8$. The probability that she passes the Physics quiz is $0.6$. The probability that she passes the Chemistry quiz is $0.75$.

(a) Find the probability that Clara passes all three quizzes.

(b) Find the probability that Clara passes exactly one of the three quizzes.

(c) Given that Clara fails the Mathematics quiz, find the probability that she passes the Physics and Chemistry quizzes.

(d) Find the probability that Clara passes at least one of the three quizzes.

A museum curator is selecting items for a new exhibit. From a collection of $20$ artifacts, $5$ are classified as ancient, and the remaining $15$ are modern. The curator randomly selects $2$ artifacts from the collection.

(a) Find the probability that exactly one of the selected artifacts is ancient.

(b) Find the probability that at least one of the selected artifacts is ancient.