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Phone Model Stock and Customer Choices

XAT 2021 · Quantitative Ability & Data Interpretation · Medium · Probability

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A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the shop to buy a unit each. Each one has a pre-determined choice of the colour and will not buy a unit of any other colour. All the three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to satisfy all the three customers?

  1. A.

    45\frac{4}{5}

  2. B.

    79\frac{7}{9}

  3. C.

    23\frac{2}{3}

  4. D.

    89\frac{8}{9}

  5. E.

    13\frac{1}{3}

Answer

C

Explanation

Total possible color choices for 3 customers = 33=273^3 = 27. Available stock: 2 White, 2 Black, 1 Red. The store fails to satisfy customers if:

  1. 3 customers choose White (needs 3, only 2 available) - 1 way (WWW)
  2. 3 customers choose Black - 1 way (BBB)
  3. 3 customers choose Red - 1 way (RRR)
  4. 2 Red and 1 White/Black - (32)×2=6\binom{3}{2} \times 2 = 6 ways (RRW, RWR, WRR, RRB, RBR, BRR) Total unsatisfied cases = 1+1+1+6=91 + 1 + 1 + 6 = 9. Satisfied cases = 27−9=1827 - 9 = 18. Probability = 1827=23\frac{18}{27} = \frac{2}{3}.

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