(a) The probabilities that three boys pass an examination are \(\frac{2}{3}, \frac{5}{8}\) and \(\frac{3}{4}\) respectively. Find the probability that : (i) all three boys pass ; (ii) none of the boyspass ; (iii) only two of the boys pass. (b) A shop-keeper marks a television set for sale at N36,000 so as to make a profit of 20% on the cost price. When he sells it, he allows a discount of 5% of the marked price. Calculate the actual percentage profit.

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(a) The probabilities that three boys pass an examination are

\frac{2}{3}, \frac{5}{8}

and

\frac{3}{4}

respectively. Find the probability that :
(i) all three boys pass ; (ii) none of the boys pass ; (iii) only two of the boys pass.
(b) A shop-keeper marks a television set for sale at N36,000 so as to make a profit of 20% on the cost price. When he sells it, he allows a discount of 5% of the marked price. Calculate the actual percentage profit.

Explanation

(a)(i) P(A) =

\frac{2}{3}

; P(B) =

\frac{5}{8}

; P(C) =

\frac{3}{4}

P (A \cap B \cap C) = P (A) \times P (B) \times P (C)

\frac{2}{3} \times \frac{5}{8} \times \frac{3}{4} = \frac{5}{16}

(ii) P(A') =

1 - \frac{2}{3} = \frac{1}{3}

P(B') =

1 - \frac{5}{8} = \frac{3}{8}

P(C') =

1 - \frac{3}{4} = \frac{1}{4}

P (A^{'} \cap B^{'} \cap C^{'}) = P (A^{'}) \times P (B^{'}) \times P (C^{'})

\frac{1}{3} \times \frac{3}{8} \times \frac{1}{4}

\frac{1}{32}

(iii) P(A and B pass, C fails) =

\frac{2}{3} \times \frac{5}{8} \times \frac{1}{4} = \frac{5}{48}

P( A and C pass, B fails) =

\frac{2}{3} \times \frac{3}{4} \times \frac{3}{8} = \frac{3}{16}

P(B and C pass, A fails) =

\frac{5}{8} \times \frac{3}{4} \times \frac{1}{3} = \frac{5}{32}

P(only two of the boys pass) =

\frac{5}{48} + \frac{3}{16} + \frac{5}{32} = \frac{43}{96}

(b) Let the cost price be x
% gain =

\frac{actual gain}{cost price} \times 100

20 = \frac{36, 000 - x}{x} \times 100

\frac{36, 000 - x}{x} = \frac{1}{5}

5 (36, 000 - x) = x ⟹ 180, 000 - 5 x = x

180, 000 = x + 5 x = 6 x

x = N 30, 000

Discount (5%) :

\frac{5}{100} \times N 36, 000 = N 1, 800

Selling price :

N (36, 000 - 1, 800) = N 34, 200

Actual gain :

\frac{34, 200 - 30, 000}{30, 000} \times 100

= 14%

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(a) The probabilities that three boys pass an examination are \(\frac{2}{3}, ...

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