(a) Two items are selected at random from four items labelled (p, q, r, s)....
(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?
Explanation
Total number of arrangements:
| p | q | r | s | |
| p | pp | pq | pr | ps |
| q | qp | qr | qs | |
| r | rp | rq | rr | rs |
| s | sp | sq | sr | ss |
(i)1. With replacement : Sample space (S) = {pp, qq, rr, ss}.
2. Without replacement : Sample space (S') = {pq, pr, ps, qr, qs, rs}
(ii)1. Total = 16; n(S) = 4 ;
\(\therefore p(S) = \frac{4}{16} = \frac{1}{4}\)
2. n(S') = 6;
\(\therefore p(S') = \frac{6}{16} = \frac{3}{8}\).
(b)(i) Numbers divisible by 4: 100, 104, 108, ..., 996.
\(T_{n} = a + (n - 1)d\)
\(996 = 100 + 4(n - 1) \implies 996 = 100 + 4n - 4\)
\(996 = 96 + 4n \implies 4n = 996 - 96 = 900\)
\(n = \frac{900}{4} = 225\)
(ii) Numbers divisible by both 3 and 4 are multiples of 12.
108, 120, 132, ..., 996.
\(996 = 108 + 12(n - 1)\)
\(996 = 108 + 12n - 12 \implies 996 = 96 + 12n\)
\(12n = 996 - 96 = 900\)
\(n = \frac{900}{12} = 75\)