Conditional probability and independent events: 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 popular cafe uses a specialty espresso machine that has an 85% probability of functioning correctly on any given morning. The probability of the machine functioning correctly on a given morning is independent of it functioning on any other morning.

Find the probability that the espresso machine functions correctly on exactly four mornings in a particular 5-day workweek.

(b) If the specialty espresso machine does not function correctly, the cafe owner must decide whether to use a backup machine or close early for the day. The cafe owner closes early if the specialty machine does not function correctly AND the backup machine is not available. There is a 70% probability that the backup machine is available on any given morning, independent of the specialty machine's status or any other morning.

Find the probability that the cafe owner does not have to close early in a particular 5-day workweek.

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.

A manufacturing plant produces electronic components. Each component undergoes two independent quality checks: Quality Check 1 (QC1) and Quality Check 2 (QC2).

The probability that a component passes QC1 is $P(QC1) = 0.8$. The probability that a component passes QC2 is $P(QC2) = 0.7$.

(a) Find the probability that a randomly selected component passes both quality checks.

(b) Find the probability that a randomly selected component passes at least one quality check.

(c) Find the probability that a randomly selected component passes Quality Check 1 but fails Quality Check 2.

(d) Find the probability that a randomly selected component passes Quality Check 2, given that it failed Quality Check 1.

[Maximum mark: 6]

A university is investigating the relationship between student engagement in extracurricular activities and their academic performance. A random sample of 470 students was selected, and their engagement level (Low, Medium, High) and academic performance (Below Average, Average, Above Average) were recorded. The data is summarized in the following table.

An item of food is chosen at random from these 500.

(a) Find the probability that a randomly chosen student has 'Below Average' academic performance, given that they have 'Low' engagement.

A $\chi^2$ test at the 5% significance level is carried out to determine if there is a significant relationship between student engagement and academic performance.

The critical value for this test is 9.488.

The hypotheses for this test are:

$H_0$: Student engagement in extracurricular activities and academic performance are independent.

$H_1$: Student engagement in extracurricular activities and academic performance are not independent.

(b) Find the $\chi^2$ statistic.

(c) State, with justification, the conclusion for this test.

The lifespan of a certain brand of smartphone battery is normally distributed with a mean of $\mu = 800$ days and a standard deviation of $\sigma = 50$ days.

(a) The probability that a randomly selected battery lasts less than $k$ days is $0.15$. Find the value of $k$.

(b) A battery is randomly selected. It is known that the battery lasts longer than $750$ days. What is the probability that it lasts longer than $820$ days?

(c) A retailer orders $200$ batteries. What is the probability that at most $62$ of them have a lifespan less than $770$ days?

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.

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 factory produces two types of electronic components: standard (S) and premium (P). The weight of these components is a critical characteristic for quality control.

The weights of standard components are known to be normally distributed with a mean of 150 grams and a standard deviation of 5 grams.

The weights of premium components are known to be normally distributed with a mean of 165 grams and a standard deviation of 8 grams.

A quality control machine classifies a component as 'premium' if its weight is found to be above 158 grams; otherwise, it is classified as 'standard'.

The factory's quality control manager uses the null hypothesis that, in the absence of other information, a component is standard.

Calculate the probability of making a Type I error when classifying a component.

Calculate the probability of making a Type II error when classifying a component.

It is known that 80% of the components produced are standard, and 20% are premium.

Calculate the overall probability that a randomly selected component is misclassified by the machine.

A manufacturing company inspects the first 50 items produced each morning for defects.

(a) State the sampling method being used.

The company uses an automated machine to test products for defects. This machine is not perfect.

It is known that 3% of all products manufactured are defective (D).

If a product is defective, the machine correctly identifies it as defective (tests positive, $T+$) 98% of the time.

If a product is not defective (D'), the machine incorrectly identifies it as defective (tests positive, $T+$) 1% of the time.

The tree diagram shows some of this information.

(b) (i) Write down the value of $P(D')$.

(ii) Write down the value of $P(T-|D)$.

(iii) Write down the value of $P(T+|D')$.

(iv) Write down the value of $P(T-|D')$.

(c) Use the tree diagram to find the probability that a randomly selected product:

(i) is not defective and tests positive.

(ii) tests negative.

(iii) is defective given that it tested negative.

(d) The company finds the actual number of defective products in their sample is different than predicted by the tree diagram. Explain why this might be the case.

(e) The factory manager surveyed all employees on a particular shift. All employees on this shift worked in at least one of these departments: Assembly (A), Painting (P), or Quality Control (Q). It was found that:

• 85 employees worked in Assembly;
• 55 employees worked in Painting;
• 35 employees worked in Quality Control;
• 10 employees worked in all three departments;
• 20 employees worked in Assembly and Quality Control but not Painting;
• 15 employees worked in Assembly and Painting but not Quality Control;
• 4 employees worked only in Quality Control.

Draw a Venn diagram to illustrate this information, placing all relevant information on the diagram.

(f) Find the total number of employees on this shift.

Leo is a student who takes a short mathematics quiz every day for five days each week. He has a 70% probability of passing any given quiz. The outcome of one quiz is independent of any other quiz.

(a) Find the probability that Leo passes exactly three quizzes in a particular five-day week.

Leo studies for his quiz with a 60% probability on any given morning, independent of whether he passes the quiz or not. Leo is considered 'well-prepared' for a quiz if he passes the quiz AND he studied for it.

(b) Find the probability that Leo is not well-prepared on any day in a particular five-day week.

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 new type of LED bulb has a lifespan, $L$, which is normally distributed with a mean of 5000 hours and a standard deviation of 300 hours.

The following curve represents this distribution. It is known that $P(L < a) = 0.1$ and $P(L > b) = 0.1$.

(a) Calculate the probability that a randomly selected LED bulb lasts more than 4700 hours.

(b) (i) Find the value of $a$.

(b) (ii) Find the value of $b$.

(c) Two LED bulbs are selected at random from a large batch. Find the probability that they both have a lifespan less than 4700 hours.

A company manufactures custom-designed shelving units. The number of shelf segments in a unit is determined by a "tier number", $N_k$. The $k$th tier number, $N_k$, represents the total number of individual shelf segments required to build a triangular display unit with $k$ levels, and is defined as $N_k = \sum_{r=1}^k r$.

Calculate the total number of shelf segments required for a unit with 6 levels, $N_6$.

Determine the formula for $N_k$ in the form $ak^2 + bk$.

A designer combines two units: one with 5 levels and another with 4 levels. Find the total number of shelf segments needed, $N_5 + N_4$.

Find the simplest expression for the total number of shelf segments required if a designer combines a unit with $k$ levels and another with $k-1$ levels, i.e., $N_k + N_{k-1}$.

In a quality control batch, there are 15 standard shelf segments and 10 reinforced shelf segments. Two segments are chosen at random from the batch without replacement.

Calculate the probability that the two segments are of different types (one standard, one reinforced).

A new batch of shelf segments contains $N_k$ standard segments and $N_{k-1}$ reinforced segments. Two segments are chosen at random from the batch without replacement.

Show that the probability that the two segments are of different types is independent of $k$.

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.

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.

The number of customers arriving at a popular bakery during peak hours can be modelled by a Poisson distribution. During a 30-minute busy period, customers arrive at a mean rate of 7.8 customers.

(a)
(i) Find the probability that exactly 6 customers arrive during a 30-minute busy period.

(a)
(ii) Find the most likely number of customers that would arrive during a 30-minute busy period.

(a)
(iii) Given that more than 8 customers arrive during a 30-minute busy period, find the probability that exactly 10 customers arrive.

(b) During quiet periods of the day, customers arrive at a mean rate of 2.1 customers every 15 minutes.

Find the probability that during a period of 45 minutes, of which the first 30 minutes is busy and the next 15 minutes is quiet, exactly 5 customers arrive.

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').

The battery life, in hours, of a particular smartphone model, $L$, can be modelled by a normal distribution with a mean of 24 hours and a standard deviation of 2 hours.

(a) Find the probability that a randomly selected smartphone has a battery life greater than 27 hours.

Two smartphones are selected at random and independently of each other.

(b) (i) Find the probability that both smartphones have a battery life greater than 27 hours.

(b) (ii) Find the probability that their total battery life is greater than 52 hours.

A software update is released which is claimed to improve battery life. The manufacturer decides to take a random sample of 20 smartphones to test this claim at the 1% significance level, assuming the standard deviation of the battery life has not changed.

(c) Write down the null and alternative hypotheses for the test.

(d) Find the critical region for this test.

Unknown to the manufacturer, the software update has resulted in all smartphones having a 5% longer battery life than the original model.

(e) Find the mean and standard deviation of the battery life for smartphones with the update.

(f) Find the probability of a Type II error in the manufacturer’s test.