Venn diagrams, tree diagrams, probability tables (+ notation & combined & mutually exclusive events): notes and practice questions

Let C be the event that a randomly chosen household in a town has a cat, and D be the event that the household has a dog. It is known that $P(C) = 0.4$, $P(D) = 0.5$ and $P(C \cap D) = 0.15$.

(a) Find the probability that a randomly chosen household has a cat or a dog.

(b) Find the probability that a randomly chosen household has neither a cat nor a dog.

A student, Chloe, travels to school by bus. She can take one of two routes, Route X or Route Y. She is equally likely to choose either route on any given day.

The probability that she arrives on time when taking Route X is $\frac{3}{4}$.

The probability that she arrives on time when taking Route Y is $\frac{2}{5}$.

(a) Find the probability that Chloe arrives on time for school on a randomly chosen day.

Let $X$ be the event that Chloe chooses Route X and let $T$ be the event that she arrives on time.

(b) Determine whether events $X$ and $T$ are independent.

(a) An electronics factory produces two types of resistors: Type A and Type B.

The resistance, $R_A$ (in Ohms), of Type A resistors is normally distributed with a mean of 120 Ohms and a standard deviation of 5 Ohms.

Find the probability that a randomly selected Type A resistor has a resistance less than 115 Ohms.

(b) In a random selection of 10 Type A resistors, find the probability that exactly 3 have a resistance less than 115 Ohms.

(c) The resistance, $R_B$ (in Ohms), of Type B resistors is normally distributed with a mean of 135 Ohms and a standard deviation of 7 Ohms.

Each day, 70% of the resistors produced are Type A, and 30% are Type B.

On a particular day, a resistor is randomly selected from all those produced at the factory.

Let $A$ represent 'Type A resistor' and $B$ represent 'Type B resistor'.

(i) Find the probability that the randomly selected resistor has a resistance less than 115 Ohms.

(ii) Given that a randomly selected resistor has a resistance less than 115 Ohms, find the probability that it is a Type A resistor.

(d) The machine that makes the Type A resistors is adjusted so that the mean resistance of the Type A resistors remains the same, but their standard deviation changes to $\sigma$ Ohms. The machine that makes the Type B resistors is not adjusted. The probability that the resistance of a randomly selected resistor from these machines is now less than 115 Ohms is 0.18.

Find the value of $\sigma$.

In a survey of a group of high school students, it was found that the probability that a student participates in the school's music program is 0.5, and the probability that a student plays on a school sports team is 0.6. The probability that a student participates in both is 0.2.

Let M be the event that a student participates in the music program and S be the event that a student plays on a sports team.

(a) Find the probability that a randomly selected student participates in the music program or plays on a sports team.

(b) Hence, find the probability that a randomly selected student participates in neither the music program nor a sports team.

In a large university, 200 students were surveyed. Of those, 120 were undergraduates (U) and the rest postgraduates (P).

Each student in the survey was asked whether they preferred quiet zones (Q) or collaborative areas (C) for studying. It was found that 75 of the undergraduates preferred quiet zones. The total number of students who preferred collaborative areas was 100. This information is shown in the following table.

Find the value of

$p$;

$q$.

Three students are chosen at random from those surveyed. Find the probability that all three are postgraduates.

Given that $P(P|Q) = \frac{1}{4}$, find the value of $x$.

A student is chosen at random from those surveyed. Write down the probability that they are a postgraduate who prefers quiet zones.

Determine if the events P (Postgraduate) and Q (prefers Quiet Zones) are independent. Justify your answer.

It can be assumed that the survey results are representative of the university population. Ten students from the university are chosen at random. Find the probability that at least five of them prefer quiet zones.

In a survey, a group of students were asked if they play football (F) or basketball (B). The probability that a student plays football is 0.6, and the probability that a student plays basketball is 0.7. The probability that a student plays at least one of these sports is 0.9.

(a) Find the probability that a student plays both football and basketball.

(b) Find the probability that a student plays neither football nor basketball.

Two archers, Clara and David, are competing. To decide who shoots an arrow, a fair six-sided die is rolled. If the die shows a 1, Clara is chosen. If the die shows any other number, David is chosen.

The probability that Clara hits the target is $\frac{4}{5}$. The probability that David hits the target is $\frac{2}{3}$.

(a) Find the probability that the target is hit.

(b) Let $C$ be the event that Clara is chosen and let $H$ be the event that the target is hit. Determine, with a reason, whether events $C$ and $H$ are independent.

(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.

Consider events $C$ and $D$ such that $P(C') = \frac{2}{3}$, $P(C \cup D) = \frac{2}{3}$ and $P(D|C) = \frac{1}{2}$.

(a) Find $P(C \cap D)$.

(b) Determine if events $C$ and $D$ are independent. Justify your answer.

At a tech company, employees can participate in various initiatives. 70% of employees attend a professional development workshop ($W$), and 20% of employees work on a special innovation project ($P$). 20% of employees do neither activity.

An employee is selected at random.

Find the probability that the employee attends a professional development workshop and works on a special innovation project.

Find the probability that the employee works on a special innovation project, but does not attend a professional development workshop.

At the company, 40% of the employees are in the Marketing department. Of those in the Marketing department, 30% work on a special innovation project.

An employee is selected at random. Let $M$ be the event "the employee is in the Marketing department" and let $P$ be the event "the employee works on a special innovation project".

Find $P(M \cap P)$.

Determine if the events $M$ and $P$ are independent. Justify your answer.

At a tech company, 60% of the employees use Python for their projects and 30% use Java. 20% of the employees use neither Python nor Java.

An employee is selected at random.

Find the probability that the employee uses both Python and Java.

Find the probability that the employee uses Java, but does not use Python.

At the company, 40% of the employees work remotely, and 35% of the remote employees use Java.

An employee is selected at random. Let R be the event "the employee works remotely" and let J be the event "the employee uses Java".

Find $P(R \cap J)$.

Determine if the events R and J are independent. Justify your answer.

(a) A fruit vendor has a basket containing $g$ green apples and $r$ red apples. Two apples are selected at random from the basket without replacement.

Let P(GG) represent the probability of selecting two green apples from the basket without replacement.

Given that P(GG) = $\frac{1}{3}$, show that $2g^2 - 2(r+1)g + r - r^2 = 0$.

(b) By solving the equation in part (a) for $g$, show that $g = \frac{(r+1)+\sqrt{3r^2 +1}}{2}$.

(c) Find two pairs of values for $r$ and $g$ that satisfy the condition P(GG) = $\frac{1}{3}$.

(d) Now consider a different basket that initially contains $10$ red apples and $g$ green apples. Three apples are selected at random from the basket without replacement.

Let P(GGG) represent the probability of selecting three green apples from the basket without replacement.

Find an expression for P(GGG) in terms of $g$.

(e) A green apple is added to the basket so that it now contains $10$ red apples and $(g+1)$ green apples.

The probability of selecting three green apples from this modified basket is now twice the probability expressed in part (d).

Find the initial number of green apples in the basket.

At a local café, the probability that a randomly chosen customer buys a coffee is $\frac{3}{5}$. The probability that they buy a pastry is $\frac{1}{2}$. The probability that they buy both a coffee and a pastry is $\frac{1}{4}$.

(a) State, with a reason, whether the events “buys a coffee” and “buys a pastry” are independent.

(b) Find the probability that a randomly chosen customer buys a coffee or a pastry or both.

(c) Find the probability that a randomly chosen customer buys exactly one of these items.

(d) Find the probability that a randomly chosen customer buys a pastry, given that they have bought a coffee.

Two fair spinners are spun. Spinner A has four equal sectors numbered 1, 2, 3, 4. Spinner B has five equal sectors numbered 1, 2, 3, 4, 5. The number on which each spinner lands is recorded.

(a) Find the probability that both spinners land on the same number.

(b) Find the probability that the sum of the two numbers is 6.

(c) Find the probability that at least one of the spinners lands on a 3.

(d) Find the probability that at least one of the spinners lands on a 4, given that the sum of the numbers is 6.

A bag contains $6$ blue marbles and $4$ yellow marbles. A marble is drawn at random from the bag and not replaced. A second marble is then drawn at random.

(a) (i) Find the probability that both marbles drawn are blue.

(ii) Find the probability that the two marbles drawn are of different colours.

A marble is now drawn at random from the bag, its colour is noted, and it is then replaced in the bag. A second marble is then drawn at random.

(b) (i) Find the probability that both marbles drawn are blue.

(ii) Find the probability that the two marbles drawn are of different colours.