Unit 10
9 marks · NormalThis question is about surds in probability calculations.
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.
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.
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(a) 8\sqrt{2}
(b) 14
(c) 2/7
(d) \frac{6}{14} \times \frac{5}{13} = \frac{15}{91}
(e) No; as long as the numbers of balls are whole numbers, the probability is always rational because it is the ratio of integers. Simplifying surds before counting means the total is always a rational count.
Common mistakes: Leaving surds unsimplified; Incorrectly calculating total balls; Not reducing probability fractions
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · Correct simplification of surds.
B1 (b) · 1 mark · Total number of balls given as 14.
M1 (c) · 1 mark · Attempts to write probability as correct fraction involving calculated totals. (follow-through)
A1 (c) · 1 mark · 2/7 given in simplest form. (follow-through)
M1 (d) · 1 mark · Attempts compound probability (multiplying fractions for both draws). (follow-through)
A1 (d) · 1 mark · First draw, second draw numbers correct (with or without ft). (follow-through)
A1 (d) · 1 mark · given as the correct, simplified answer. (follow-through)
B1 (e) · 1 mark · Clear, correct explanation referencing ratio of integers or surd simplification.
B1 (e) · 1 mark · Concludes that probability must be rational if counts are whole numbers.
Worked solution
(a)
(b) . So, number of red balls = 6, blue = 8; total = 14.
(c) Probability of blue: (OR, blue balls = 8 out of 14).
But the question asks for simplest form (also accept 8/14).
(d) For both red: First draw: 6/14, Second draw (one red gone, 13 left): 5/13.
(e) The number of balls is always a whole number so the probability is always the ratio of integers. Even if surds are given, once simplified, the count is rational, so probabilities here are rational.