Conditional probability

Sam has 9 cards, each with a different non-zero digit printed on it.

Sam chooses 5 cards at random and places them in a random order in a straight line, to form a 5-digit number.

Find the probability that the 5-digit number is greater than 50 000.

Given that the 5-digit number is greater than 50 000, determine the probability that it is

greater than 58 600.

In this question you must show detailed reasoning.

A disease that affects trees shows no visible evidence for the first few years after the tree is infected.

A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model.

• If the tree has the disease, the probability of a positive result is 0.95.

• If the tree does not have the disease, the probability of a positive result is 0.1.

It is known that in a certain county, A, 35% of the trees have the disease. A tree in county A is chosen at random and is tested.

Given that the result is positive, determine the probability that this tree has the disease.

A forestry company wants to determine what proportion of trees in another county, B, have the disease. They choose a large random sample of trees in county B.

Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.

By carrying out a calculation, determine an estimate of the proportion of trees in county B

that have the disease.

There are 25 students in a class.

• The number of students who study both History and English is 3.

• The number of students who study neither History nor English is 14.

• The number of students who study History but not English is three times the number who study English but not History.

• Show this information on a Venn diagram.

• Determine the probability that a student selected at random studies English.

Two different students from the class are chosen at random.

Given that exactly one of the two students studies English, determine the probability that

exactly one of the two students studies History.

Anika and Beth are playing a game which consists of several points.

• The probability that Anika will win any point is 0.7.

• The probability that Beth will win any point is 0.3.

• The outcome of each point is independent of the outcome of every other point.

The first player to win two points wins the game.

Write down the probability that the game consists of more than three points.

Complete the probability tree diagram in the Printed Answer Booklet showing all the

possibilities for the game.

Determine the probability that Beth wins the game.

Determine the probability that the game consists of exactly three points.

Given that Beth wins the game, determine the probability that the game consists of exactly

three points.

The values of three independent observations of X are denoted by X₁, X₂ and X₃.

Given that X₁ + X₂ + X₃ = 9, determine the probability that at least one of these three values is

equal to 2.

Freda chooses values of X at random until she has obtained X = 2 exactly three times. She then stops.

Determine the probability that she chooses exactly 10 values of X.

Find the probability that S is odd.

Find the probability that S is greater than 8, given that S is odd.

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable Y defined as follows. P(Y = y+1) = (Y = y) for all positive integers y.

Find P(Y = 1).

Give a reason why one of the variables, X or Y, might be more appropriate as a model for the number

of attempts that Sheila needs to start her car.