Probability

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.

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).

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.

The discrete random variable X has probability function

Find P(X = 2) and complete the table below.

Given that E(X) = 1.6

Find the value of a and the value of b.

Find

P(0.5 < X < 3),

E(3 X – 2).

Show that the Var(X) = 1.64

Calculate Var(3 X – 2).

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.