Mutually exclusive & independent events

The records for a school athletics club show that the height, H metres, achieved by students in the high jump is normally distributed with mean 1.4 metres and standard deviation 0.15 metres.

Find the proportion of these students achieving a height of more than 1.6 metres. The records also show that the time, T seconds, to run 1500 metres is normally distributed with mean 330 seconds and standard deviation 26 seconds. The school’s Head would like to use these distributions to estimate the proportion of students from the school athletics club who can jump higher than 1.6 metres and can run 1500 metres in less than 5 minutes.

State a necessary assumption about H and T for the Head to calculate an estimate of this proportion.

Find the Head’s estimate of this proportion. Students in the school athletics club also throw the discus.

Given that P(D 16.3) = 0.30 and P(D 29.0) = 0.10

calculate the value of µ and the value of σ

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 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

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.

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.

A biased die with six faces is rolled. The discrete random variable X represents the score on the uppermost face. The probability distribution of X is shown in the table below.

Given that E(X) = 4.2 find the value of a and the value of b.

Show that E(X²) = 20.4

Find Var(5 – 3 X)

A biased die with five faces is rolled. The discrete random variable Y represents the score which is uppermost. The cumulative distribution function of Y is shown in the table

Find the value of k.

Find the probability distribution of Y.

Each die is rolled once. The scores on the two dice are independent.

Find the probability that the sum of the two scores equals 2

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.

The score S when a spinner is spun has the following probability distribution.

Find E(S).

Show that E(S²) = 10.4

Hence find Var(S).

Find

E(5 S – 3),

Var(5 S – 3).

Find P(5 S – 3 > S + 3)

The spinner is spun twice. The score from the first spin is S₁ and the score from the second spin is S₂ The random variables S₁ and S₂ are independent and the random variable X = S₁ × S₂

Show that P({S₁ = 1} X < 5) = 0.16

Find P(X < 5).

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).

A spinner is designed so that the score S is given by the following probability distribution.

Find the value of p.

Find E(S).

Find Var(S).

Tom and Jess play a game with this spinner. The spinner is spun repeatedly and S counters are awarded on the outcome of each spin. If S is even then Tom receives the counters and if S is odd then Jess receives them. The first player to collect 10 or more counters is the winner.

Find the probability that Jess wins after 2 spins.

Find the probability that Tom wins after exactly 3 spins.

Find the probability that Jess wins after exactly 3 spins.

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 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.