Conditional probability

Bag A contains 5 red, 4 yellow and 3 green beads. Bag B only contains red and yellow beads. A bead is selected at random from bag A and a second bead is selected at random from bag B Given that the probability that both beads selected are yellow is

find the probability of selecting a yellow bead from bag B

Hence complete the tree diagram below by finding the probability on each branch.

Find the exact probability that at least one of the two beads selected is yellow. The event X is that at least one of the beads selected is yellow. The event W is that a green bead is selected.

Find the exact value of P(W|X)

The Venn diagram, where p, q and r are probabilities, shows the events A, B, C and D and associated probabilities.

State any pair of mutually exclusive events from A, B, C and D The events B and C are independent.

Find the value of p

Find the greatest possible value of P A | B Given that P B | A 0 5.

find the value of q and the value of r

Find P A B C

Use set notation to write an expression for the event with probability p

TOTAL FOR STATISTICS IS 50 MARKS

The Venn diagram, where p and q are probabilities, shows the three events A, B and C and their associated probabilities.

Find P(A) The events B and C are independent.

Find the value of p and the value of q

Find P(AB′)

A company has 1825 employees. The employees are classified as professional, skilled or elementary. The following table shows

• the number of employees in each classification

• the two areas, A or B, where the employees live

An employee is chosen at random. Find the probability that this employee

is skilled,

lives in area B and is not a professional. Some classifications of employees are more likely to work from home.

• 65% of professional employees in both area A and area B work from home

• 40% of skilled employees in both area A and area B work from home

• 5% of elementary employees in both area A and area B work from home

• Event F is that the employee is a professional

• Event H is that the employee works from home

• Event R is that the employee is from area A

Using this information, complete the Venn diagram on the opposite page.

Find P (R' F)

Find P([H R]')

Find P (F | H)

Turn over for a spare diagram if you need to redraw your Venn diagram.

Only use this diagram if you need to redraw your Venn diagram.

State one disadvantage of using quota sampling compared with simple random sampling. In a university 8% of students are members of the university dance club. A random sample of 36 students is taken from the university. The random variable X represents the number of these students who are members of the dance club.

Using a suitable model for X, find

P(X = 4)

P(X 7) Only 40% of the university dance club members can dance the tango.

Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.

Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango.

A large college produces three magazines. One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events G, E and S are defined as follows G is the event that the student reads the magazine about green issues E is the event that the student reads the magazine about equality S is the event that the student reads the magazine about sports The Venn diagram, where p, q, r and t are probabilities, gives the probability for each

Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and P(G) = 0.25

Find

the value of p

the value of q

find

the value of r

the value of t

Determine whether or not the events (S Eʹ) and G are independent. Show your working clearly.

The Venn diagram shows the probabilities associated with four events, A, B, C and D

Write down any pair of mutually exclusive events from A, B, C and D Given that P(B) = 0.4

find the value of p Given also that A and B are independent

find the value of q Given further that P(Bʹ| C) = 0.64

find

the value of r

the value of s

A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.

Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.

Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients’ complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, T minutes, can be modelled by the normal distribution where T~ N(5, 3.5²)

Using this model,

find the probability that a routine appointment with the dentist takes less than 2 minutes

find P(T < 2 | T 0)

hence explain why this normal distribution may not be a good model for T. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of T 2

Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.

TOTAL FOR STATISTICS IS 50 MARKS

A bag contains 64 coloured beads. There are r red beads, y yellow beads and 1 green bead and r + y + 1 = 64 Two beads are selected at random, one at a time without replacement.

Find the probability that the green bead is one of the beads selected. The probability that both of the beads are red is

Show that r satisfies the equation r² – r – 240 = 0

Hence show that the only possible value of r is 16

Given that at least one of the beads is red, find the probability that they are both red.

The Venn diagram shows three events A, B and C, where p, q, r, s and t are probabilities.

P(A) = 0.5, P(B) = 0.6 and P(C) = 0.25 and the events B and C are independent.

Find the value of p and the value of q.

Find the value of r.

Hence write down the value of s and the value of t.

State, giving a reason, whether or not the events A and B are independent.

Yuto works in the quality control department of a large company. The time, T minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.

Find the probability that Yuto takes longer than 20 minutes to analyse the next sample.

The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.

Find the probability that this sample took Yuto more than 20 minutes to analyse.

Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.

Estimate the median time taken by Yuto to analyse samples in future.

The discrete random variable X has the following probability distribution, where p and q are constants.

Write down an equation in p and q

Given that E(X) = 0.4

find the value of q

hence find the value of p

Given also that E(X²) = 2.275

find Var(X)

Sarah and Rebecca play a game. A computer selects a single value of X using the probability distribution above. Sarah’s score is given by the random variable S = X and Rebecca’s score is given by the

Find E(R)

Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.

Find the probability that

Sarah is the winner,

Rebecca is the winner.

The Venn diagram shows the probabilities of customer bookings at Harry’s hotel. R is the event that a customer books a room B is the event that a customer books breakfast D is the event that a customer books dinner u and t are probabilities.

Write down the probability that a customer books breakfast but does not book a room.

Given that the events B and D are independent

find the value of t

hence find the value of u

Find

P(DR B)

A coach load of 77 customers arrive at Harry’s hotel. Of these 77 customers 40 have booked a room and breakfast 37 have booked a room without breakfast

Estimate how many of these 77 customers will book dinner.

A college has 80 students in Year 12. 20 students study Biology 28 students study Chemistry 30 students study Physics 7 students study both Biology and Chemistry 11 students study both Chemistry and Physics 5 students study both Physics and Biology 3 students study all 3 of these subjects

Draw a Venn diagram to represent this information.

A Year 12 student at the college is selected at random.

Find the probability that the student studies Chemistry but not Biology or Physics.

Find the probability that the student studies Chemistry or Physics or both.

Given that the student studies Chemistry or Physics or both,

find the probability that the student does not study Biology.

Determine whether studying Biology and studying Chemistry are statistically independent.

In a company the 200 employees are classified as full-time workers, part-time workers or contractors. The table below shows the number of employees in each category and whether they walk to work or use some form of transport.

The events F, H and C are that an employee is a full-time worker, part-time worker or contractor respectively. Let W be the event that an employee walks to work. An employee is selected at random. Find

P(H)

P(W C)

Let B be the event that an employee uses the bus. Given that 10% of full-time workers use the bus, 30% of part-time workers use the bus and 20% of contractors use the bus,

draw a Venn diagram to represent the events F, H, C and B,

find the probability that a randomly selected employee uses the bus to travel to work.

Figure 1

The Venn diagram in Figure 1 shows three events A, B and C and the probabilities associated with each region of B. The constants p, q and r each represent probabilities associated with the three separate regions outside B. The events A and B are independent.

Find the value of p.

find the value of q and the value of r.

A fair blue die has faces numbered 1, 1, 3, 3, 5 and 5. The random variable B represents the score when the blue die is rolled.

Write down the probability distribution for B.

State the name of this probability distribution.

Write down the value of E(B).

A second die is red and the random variable R represents the score when the red die is rolled. The probability distribution of R is

Find E(R).

Find Var(R).

Tom invites Avisha to play a game with these dice. Tom spins a fair coin with one side labelled 2 and the other side labelled 5. When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.

Find the probability that Avisha wins the game, stating clearly which die she should use in each case.

Given that P(A) = 0.35, P(B) = 0.45 and P(A B find

P()

The event C has P(C) = 0.20 The events A and C are mutually exclusive and the events B and C are independent.

Draw a Venn diagram to illustrate the events A, B and C and the probabilities for each region.

Figure 1 shows how 25 people travelled to work. Their travel to work is represented by the events B bicycle T train W walk

Write down 2 of these events that are mutually exclusive. Give a reason for your answer.

Determine whether or not B and T are independent events.

One person is chosen at random. Find the probability that this person

walks to work,

travels to work by bicycle and train.

Given that this person travels to work by bicycle, find the probability that they will also take the train.

A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A toy without poor stitching has a probability of 0.02 of splitting open.

Draw a tree diagram to represent this information.

Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching or splitting open.

The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is independent of poor stitching or splitting open. A soft toy is chosen at random.

Find the probability that the soft toy has none of these 3 defects.

Find the probability that the soft toy has exactly one of these 3 defects.

Jake and Kamil are sometimes late for school. The events J and K are defined as follows J = the event that Jake is late for school K = the event that Kamil is late for school P(J) = 0.25, P(J K) = 0.15 and P(J K) = 0.7 On a randomly selected day, find the probability that

at least one of Jake or Kamil are late for school,

Kamil is late for school.

Given that Jake is late for school,

find the probability that Kamil is late.

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

Determine whether or not J and K are statistically independent.

Comment on the teacher’s suspicion in the light of your calculation in (d).

The bag P contains 6 balls of which 3 are red and 3 are yellow. The bag Q contains 7 balls of which 4 are red and 3 are yellow. A ball is drawn at random from bag P and placed in bag Q. A second ball is drawn at random from bag P and placed in bag Q. A third ball is then drawn at random from the 9 balls in bag Q. The event A occurs when the 2 balls drawn from bag P are of the same colour. The event B occurs when the ball drawn from bag Q is red.

Complete the tree diagram shown below.

Find P(A)

Given that all three balls drawn are the same colour, find the probability that they are all red.

An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability of landing heads is spun. When a blue ball is selected a fair coin is spun.

Complete the tree diagram below to show the possible outcomes and associated probabilities. Ball Coin

Shivani selects a ball and spins the appropriate coin.

Find the probability that she obtains a head.

Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,

find the probability that Tom selected a red ball.

Shivani and Tom each repeat this experiment.

Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

The discrete random variable X has probability distribution given by

x	1	0	1	2	3
P(X x)	1 5	a	1 10	a	1 5

where a is a constant.

Find the value of a.

Write down E(X).

Find Var(X).

Find Var(Y).

The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines A, B and C.

One of these students is selected at random.

Show that the probability that the student reads more than one magazine is

Find the probability that the student reads A or B (or both).

Write down the probability that the student reads both A and C.

Given that the student reads at least one of the magazines,

find the probability that the student reads C.

Determine whether or not reading magazine B and reading magazine C are statistically independent.

On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is , and respectively. The probability of being late when using these methods of travel is and respectively.

Draw a tree diagram to represent this information.

Find the probability that on a randomly chosen day

Bill travels by foot and is late,

Bill is not late.

Given that Bill is late, find the probability that he did not travel on foot.

Given that P(A) = a and P(B) = b express P(A B) in terms of a and b when

A and B are mutually exclusive,

A and B are independent.

Two events R and Q are such that P(R Q ) = 0.15, P(Q) = 0.35 and P(R|Q) = 0.1 Find the value of

P(R).

A group of office workers were questioned for a health magazine and were found to take regular exercise. When questioned about their eating habits said they always eat breakfast and, of those who always eat breakfast also took regular exercise. Find the probability that a randomly selected member of the group

always eats breakfast and takes regular exercise,

does not always eat breakfast and does not take regular exercise.

Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.

A disease is known to be present in 2% of a population. A test is developed to help determine whether or not someone has the disease. Given that a person has the disease, the test is positive with probability 0.95 Given that a person does not have the disease, the test is positive with probability 0.03

Draw a tree diagram to represent this information.

A person is selected at random from the population and tested for this disease.

Find the probability that the test is positive.

A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive,

find the probability that he does not have the disease.

Comment on the usefulness of this test.

A person’s blood group is determined by whether or not it contains any of 3 substances A, B and C. A doctor surveyed 300 patients’ blood and produced the table below.

Draw a Venn diagram to represent this information.

Find the probability that a randomly chosen patient’s blood contains substance C.

Harry is one of the patients. Given that his blood contains substance A,

find the probability that his blood contains all 3 substances.

Patients whose blood contains none of these substances are called universal blood donors.

Find the probability that a randomly chosen patient is a universal blood donor.

The following shows the results of a wine tasting survey of 100 people. 96 like wine A, 93 like wine B, 96 like wine C, 92 like A and B, 91 like B and C, 93 like A and C, 90 like all three wines.

Draw a Venn Diagram to represent these data.

Find the probability that a randomly selected person from the survey likes

none of the three wines,

wine A but not wine B,

any wine in the survey except wine C,

exactly two of the three kinds of wine.

Given that a person from the survey likes wine A,

find the probability that the person likes wine C.

Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable R is the score on the red die and the random variable B is the score on the blue die.

Find P(R =3 and B =0).

The random variable T is R multiplied by B.

Complete the diagram below to represent the sample space that shows all the possible values of T.

The table below represents the probability distribution of the random variable T.

Find the values of a, b, c and d.

Find the values of

E(T),

Var(T).