Unit 10
9 marks · NormalThis problem links rational and irrational numbers with probability.
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.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) Rational; can be written as 1/3
(b) Impossible; sum of two integers is always integer (rational), never irrational; so probability is 0
(c) Always rational; product of two integers is always integer (rational), so probability is 1
(d) The square root is irrational unless the product is a perfect square. List all red * blue pairs: only when the red die result matches the square root of the blue number does the product give a perfect square (e.g. red=4, blue=4: 16; red=3, blue=9: 27 etc). In all other cases, sqrt(product) is irrational. There are 6 × 6 = 36 possible outcomes. Only 6 (when red die is the square root of blue die) give a rational result. So, probability is (36-6)/36 = 30/36 = 5/6.
Common mistakes: Mistaking recurring decimals as irrational; Ignoring impossibility of irrational integer sums
Show mark scheme and worked solution
Marking points
B1 (a) · 2 marks · States recurring decimal is rational, correct explanation.
B1 (b) · 2 marks · Correct argument and answer: probability 0 with reasoning.
B1 (c) · 2 marks · Product of two integers is rational, so probability 1.
M1 (d) · 1 mark · Identifies which products give a rational square root. (follow-through)
M1 (d) · 1 mark · Counts correct number of outcomes for rational/irrational square roots. (follow-through)
A1 (d) · 1 mark · Probability as 5/6. (follow-through)
Worked solution
(a) 0.333... is 1/3, which is a ratio of integers, so it is rational.
(b) Sums on two dice are integers; an integer cannot be irrational; so probability is 0.
(c) Product of two integers is always rational.
(d) The product is a perfect square if the red die result is a square root of the blue die number: (1,1), (2,4), (3,9), (4,16), (5,25), (6,36) → 6 outcomes. For the other 30 outcomes, the root is irrational. So probability is 30/36 = 5/6.