(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?
(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?
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Correct Answer: Option n
Explanation:
Total number of arrangements:
(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.
\(996 = 108 + 12(n - 1)\)
\(996 = 108 + 12n - 12 \implies 996 = 96 + 12n\)
\(12n = 996 - 96 = 900\)
\(n = \frac{900}{12} = 75\)
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.
\(996 = 108 + 12(n - 1)\)
\(996 = 108 + 12n - 12 \implies 996 = 96 + 12n\)
\(12n = 996 - 96 = 900\)
\(n = \frac{900}{12} = 75\)