NCEA Level 3 Statistics 91585 Probability free practice questions with answers

Question 1 (14 marks) [Two-way tables and Venn diagrams]

(a) A survey of 100 households surveyed recorded whether each person have solar panels and whether they use a heat pump. The results are in the table. (i) Calculate the probability that a randomly chosen person have solar panels and use a heat pump. (ii) Calculate the probability that a person chosen at random from those who use a heat pump also have solar panels, and compare it with the probability that a person chosen at random from everyone surveyed have solar panels. (iii) Decide whether ‘have solar panels’ and ‘use a heat pump’ are independent, justifying your decision with probabilities, and explain what your answer means in this context.

(a) Answer: (i)

• Evidence level: Merit (read the cells of the two-way table, then compare P(A | B) with P(A)).

• (i) P(have solar panels and use a heat pump) = 26 ÷ 100 =

• (ii) P(have solar panels | use a heat pump) = 26 ÷ 59 = 100 =

• Achievement: (iii) States whether the events are not independent.

• Merit: Links to the probabilities: independent events have P(A | B) = P(A); here they are different.

• Excellence: Full explanation: the events are not independent because P(have solar panels | use a heat pump) = is more likely to have solar panels than a person chosen from everyone (the events are associated).

(b) Of 150 members of a tramping club, 65 own a tent, 27 own a stove, and 15 do both. The rest do neither. (i) Draw a Venn diagram showing the number in each region. (ii) Calculate the probability that a randomly chosen member does own a tent or own a stove (or both), and the probability that they do neither. (iii) Calculate P(own a tent | own a stove). Decide whether the events are mutually exclusive, and whether they are independent, justifying each answer with probabilities.

(b) Answer: (ii)

• Evidence level: Merit (fill the Venn regions from the middle outwards; use P(A or B) = P(A) + P(B) − P(A and B)).

• (i) Both: 15; own a tent only: 65 − 15 = 50; own a stove only: 27 − 15 = 12; neither: 150 − 15 − 50 − 12 = 73.

• (ii) P(own a tent or own a stove) = (50 + 15 + 12) ÷ 150 = 77 ÷ 150 =

• (iii) P(own a tent | own a stove) = 15 ÷ 27 = Independent? P(own a tent) × P(own a stove) = are not independent. 13 50 ; (ii) 26 59 compared with 37 100 ; (iii) not independent 13 50 = 0.260. 26 59 = 0.441; P(have solar panels) = 37 ÷ 37 100 = 0.370. 26 59 ≠ P(have solar panels) = 37 100 ; this means that a person who use a heat pump 77 150 and 73 150 ; (iii) 5 9 ; not mutually exclusive; not independent 77 150 ; P(neither) = 73 ÷ 150 = 73 150 . 5 9 . Mutually exclusive? No: P(both) = 1 10 is not zero. 13 30 × 9 50 = 39 500 ≠ P(both) = 1 10 , so the events