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WAEC 2018 · Paper 2 · Q13

One out of every 3 bolts produced by a machine is defective. If 4 of the bolts produced by the machine are selected at random, find the probability that:

  1. (a)

    exactly 2 are defective;

  2. (b)

    at least 1 is defective;

  3. (c)

    at most 2 are defective.

Worked solution (try it first)
  1. X∼B(4,13)X \sim B\left(4, \frac13\right), with q=23q = \frac23.

(a)

  1. P(X=2)P(X = 2)
    =(42)(13)2(23)2= \binom42\left(\frac13\right)^2\left(\frac23\right)^2
    =2481= \dfrac{24}{81}
    =827= \dfrac{8}{27}.

(b)

  1. At least 1: 1−(23)4=1−16811 - \left(\frac23\right)^4 = 1 - \dfrac{16}{81}
    =6581= \dfrac{65}{81}.

(c)

  1. At most 2: P(0)+P(1)+P(2)=16+32+2481P(0) + P(1) + P(2) = \dfrac{16 + 32 + 24}{81}
    =7281= \dfrac{72}{81}
    =89= \dfrac89.

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