Maths · Probability · chapter 13 of 13
Class 12 Maths Chapter 13: Probability case study question with answers
One case-based question of the kind the board asks: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.
Case study · 4 marks
A cab app in Ahmedabad sends either a Gold driver or a Silver driver to each booking. Over a month, 60% of bookings went to Gold drivers and 40% to Silver drivers. A Gold driver reached the pick-up point within 5 minutes on 90% of bookings, a Silver driver on 60%. A booking is chosen at random from the month's records. Let G be the event 'a Gold driver was sent' and W the event 'the cab arrived within 5 minutes'.
Part 1 · 1 mark
P(W) is
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Answer: (c) 0.78
G and G′ form a partition: P(W) = 0.6 × 0.9 + 0.4 × 0.6 = 0.54 + 0.24 = 0.78. (b) is P(G ∩ W) alone; (a) averages 0.9 and 0.6.
Part 2 · 1 mark
Given that the cab did not arrive within 5 minutes, the probability that a Silver driver was sent is
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Answer: (a) 811
P(W′) = 0.22 and P(G′ ∩ W′) = 0.4 × 0.4 = 0.16, so P(G′|W′) = 0.16/0.22 = 8/11. (c) is P(G|W′); (b) is the a priori probability.
Part 3 · 2 marks
Given that the cab arrived within 5 minutes, find the probability that a Gold driver was sent.
Answer and marking scheme
By Bayes' theorem, P(G|W) = P(G) P(W|G)/P(W) = 0.54/0.78 = 9/13.
- P(G ∩ W) = 0.54 over P(W) = 0.78 (1)
- 9/13 (1)
Or
Are the events G and W independent? Justify your answer.
Answer and marking scheme
P(G ∩ W) = 0.54, while P(G) P(W) = 0.6 × 0.78 = 0.468. Since 0.54 ≠ 0.468, G and W are not independent (equally, P(W|G) = 0.9 ≠ P(W) = 0.78).
- P(G ∩ W) = 0.54 and P(G) P(W) = 0.468 (1)
- unequal, so not independent (1)