Unit 10

Simplify . Give your answer as a surd in its simplest form.

A bag contains 6 red balls and blue balls. A ball is chosen at random.

i) Write the total number of balls in the bag as a rational number.

ii) Write down, as a fraction in its simplest form, the probability of choosing a blue ball.

A second ball is chosen from the bag without replacement. Calculate the probability that both balls selected are red. Give your answer as a fraction in its simplest form.

Is it possible to get an irrational probability from drawing balls from this bag? Explain your answer clearly.

Sets , and are defined as follows:

List all the elements of sets , , , and .

Represent the sets , and in a Venn diagram. Clearly indicate and label the region .

A number is selected at random from set .

Draw a tree diagram to show the possible outcomes and probabilities for 'multiple of 2' and 'multiple of 3' in set .

Calculate the probability that a number chosen at random from set is both a multiple of 2 and a multiple of 3.

Using your tree diagram or otherwise, calculate the probability that the number chosen is a multiple of 2 or a multiple of 3 (or both). Give your answer as a fraction in its simplest form.

Leo is working with the expression . Decide whether is rational or irrational. Explain your answer.

A numbered spinner has 5 sections: 2 labelled 'A' and 3 labelled 'B'. Each time it is spun, the outcome is equally likely.

If the spinner is spun twice, find, with reasons, the probability of getting 'A' on both spins. Write your answer as a fraction in its lowest terms.

Expand the expression and comment on whether the result is rational or irrational.

The spinner is spun twice. Given that 'A' appeared at least once, what is the probability that the first spin was 'A'? Give your reasoning and write your answer as a fraction in its simplest form.

State, with a reason, whether the decimal is rational or irrational.

A die has faces numbered . Find the probability that two dice, both thrown once, will show numbers that sum to an irrational number. Explain your answer.

Now suppose a red die is numbered 1, 2, 3, 4, 5, 6, and a blue die is numbered 1, 4, 9, 16, 25, 36. A red and blue die are rolled together and the product of the two numbers is recorded.

i) Find the probability that the product is a rational number.

ii) The dice are rolled again. Find the probability the square root of their product is irrational.

Express the following as a single fraction in its simplest form:

Given that 8 alarms were installed that month, solve the equation above to find the cost per alarm, c.

A follow-up promotion means that the number of alarms in a month is now a random variable with the following probabilities:

A supervisor picks a month at random from the coming year and uses the value of n from that month in the same cost equation as above, with c = 12. (i) For each possible value of n, state if the equation has a solution for n > 0. Briefly explain your answer.

The supervisor now wants to know, "What is the probability that the equation has a valid solution for n > 0, based on the probability distribution in part b?"

Complete a probability tree diagram for randomly selecting a value of n and checking if the equation can be solved for n > 0. Clearly label the branches with numbers and probabilities.

Use your tree diagram to find the probability that a randomly selected month leads to a valid value of n from part (c).

Solve the equation for x, giving all integer solutions.

Given that x must be a possible die score, what is the probability that Lia's roll will give a solution to the equation?

In another round, Lia rolls the die twice. She finds the solution to the equation for each value of x, and records whether the equation is solvable (using only possible die scores).

Draw a tree diagram to show all possible outcomes and the probabilities for both rolls, where S means the equation is solvable for the roll, and N means it is not solvable.

Use your diagram to calculate the probability that exactly one of the two rolls gave a score making the equation solvable.

Simplify the following expression, giving your answer in its simplest form:

A bag contains only red and blue counters. The probability that a counter selected at random is red is given by

where .

Two counters are selected from the bag, one after the other, with replacement.

i) Draw a tree diagram to show all possible outcomes for the colours of the two counters. Label each branch clearly with its probability in terms of .

ii) Express, in terms of , the probability that exactly one of the two counters selected is red.

If the probability that exactly one of the counters chosen is red is , find the value of .