Unit 10
10 marks · HardLia rolls a fair six-sided die. The probability of rolling each number is .
Part of her scoring game is described by the equation:
where is the score Lia rolls.
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.
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) x = 1 or x = -2, but only x = 1 is a possible die score.
(b) P(x = 1) = 1/6; Probability = 1/6
(c) Tree diagram with first and second roll, S/N for each, showing probabilities (1/6 for S, 5/6 for N on each roll). · must show Correctly labelled branches for S/N and probabilities for both rolls (first and second)
(d) Probability = 2 × (1/6) × (5/6) = 10/36 = 5/18 ≈ 0.278
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Sets up and rearranges the equation correctly.
A1 (a) · 2 marks · Finds all integer solutions and correctly identifies which are possible on a die.
M2 (b) · 1 mark · Identifies die scores possible and links to probability.
A2 (b) · 1 mark · Correct calculation of probability: 1/6.
M3 (c) · 1 mark · Draws tree diagram with two levels (first and second roll), each with S/N branches and probabilities.
A3 (c) · 2 marks · Correctly calculates all branch probabilities and fully labels tree.
M4 (d) · 1 mark · Selects correct branches for 'exactly one S' outcome. (follow-through)
A4 (d) · 1 mark · Adds probabilities to get 2 × (1/6) × (5/6) = 5/18 ≈ 0.278. (follow-through)
Worked solution
a) Set up the equation:
Multiply both sides by (x² - 1):
Factorise:
Only positive integer values for a die: x = 1 is possible.
b) The die can show any of 1–6. Only x = 1 gives a solution. P(solution) = 1/6 = 0.1667.
c) Tree diagram for two rolls: First roll (S: solvable, N: not solvable): - S (x=1): 1/6 - N (x=2,3,4,5,6): 5/6 Second roll (independent, same probabilities): - S (1/6), N (5/6) from each previous branch. Branches: S/S, S/N, N/S, N/N
d) Probability that exactly one roll is S: Two cases: S then N, or N then S, each probability (1/6)*(5/6), total probability 2*(1/6)*(5/6) = 10/36 = 5/18 ≈ 0.278