(a) Two items are selected at random from four... | WAEC SSCE - Further Mathematics - 2012 (Essay)

Question 1 Report

(a) Two items are selected at random from four items labelled (p, q, r, s).

(i) List the sample space if sampling is done (1) with replacement ; (2) without replacement.

(ii) Find the probability that r is at least one of the two objects selected : (1) in a(i)1 ; (2) in a(i)2.

(b) How many whole numbers from 100 to 999 are divisible by (i) 4 ; (ii) both 3 and 4?

Answer

(a)(i)(1) With replacement (ordered pairs, \(4\times4 = 16\)):

\(pp, pq, pr, ps,\ qp, qq, qr, qs,\ rp, rq, rr, rs,\ sp, sq, sr, ss\).

(2) Without replacement (ordered pairs, \(4\times3 = 12\)):

\(pq, pr, ps,\ qp, qr, qs,\ rp, rq, rs,\ sp, sq, sr\).

(ii) "\(r\) is at least one of the two."

(1) Outcomes with no \(r\): \(3\times3 = 9\), so outcomes containing \(r = 16 - 9 = 7\).

\[P(r) = \frac{7}{16}\]

(2) Outcomes with no \(r\): \(3\times2 = 6\), so outcomes containing \(r = 12 - 6 = 6\).

\[P(r) = \frac{6}{12} = \frac{1}{2}\]

(b) Whole numbers from \(100\) to \(999\).

(i) Divisible by 4: from \(100\ (=4\times25)\) to \(996\ (=4\times249)\):

\[249 - 25 + 1 = 225\ \text{numbers}\]

(ii) Divisible by both 3 and 4 (i.e. by 12): from \(108\ (=12\times9)\) to \(996\ (=12\times83)\):

\[83 - 9 + 1 = 75\ \text{numbers}\]

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