Conditional probability

Three events X, Y and Z are such that

P(X) = P(Y) = 0.38 P(X ∪ Z) = 0.5 P [ (X ∪ Y) ' ] = 0.35 X and Z are mutually exclusive.

Find P(Z)

Complete the Venn diagram.

X Y Z

0.27........

0.3

Find P[(X ∩ Y) ∪ (Y ∩ Z)]

Find P(Y' | Z')

Show that the events Y and Z are not independent of each other.

There are two types of coins in a money box:

20% are bronze coins

80% are silver coins

Craig takes out a coin at random and places it back in the money box.

Craig then takes out a second coin at random.

Find the probability that both coins were of the same type.

Find the probability that both coins are bronze, given that at least one of the coins is bronze.

A sample of 240 households were asked which, if any, of the following animals they own as pets:

The results are shown in the table below.

Represent this information by fully completing the Venn diagram below.

A household is chosen at random from the sample.

Find the probability that the household owns a cat only.

Find the probability that the household owns at least two of the three types of pet.

Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise.

Determine whether a household owning a cat and a household owning a tortoise are independent of each other.

Fully justify your answer.

A and B are two events such that

Find P(A)

Find P(BA)

Determine if A and B are independent events.

The probabilities of events A, B and C are related, as shown in the Venn diagram below.

Find the value of x.

Circle your answer.

0.11 0.46 0.54 0.89

Diedre is a head teacher in a school which provides primary, secondary and sixth-form education.

There are 200 teachers in her school.

The number of teachers in each level of education along with their gender is shown in the table below.

A teacher is selected at random. Find the probability that:

the teacher is female

the teacher is not a sixth-form teacher.

Given that a randomly chosen teacher is male, find the probability that this teacher is not a primary teacher.

Diedre wants to select three different teachers at random to be part of a school project.

Calculate the probability that all three chosen are secondary teachers.

A survey was conducted into the health of 120 teachers.

The survey recorded whether or not they had suffered from a range of four health issues in the past year.

In addition, their physical exercise level was categorised as low, medium or high.

50 teachers had a low exercise level, 40 teachers had a medium exercise level and 30 teachers had a high exercise level.

The results of the survey are shown in the table below.

Find the probability that a randomly selected teacher:

suffers from back trouble and has a high exercise level;

suffers from depression.

suffers from stress, given that they have a low exercise level.

For teachers in the survey with a low exercise level, explain why the events ‘suffers from back trouble’ and ‘suffers from stress’ are not mutually exclusive.

A teacher in a college asks her mathematics students what other subjects they are studying.

She finds that, of her 24 students:

12 study physics 8 study geography 4 study geography and physics

A student is chosen at random from the class.

Determine whether the event ‘the student studies physics’ and the event ‘the student studies geography’ are independent.

It is known that for the whole college:

the probability of a student studying mathematics is

the probability of a student studying biology is

the probability of a student studying biology given that they study mathematics is

Calculate the probability that a student studies mathematics or biology or both.