(a) Eight coins are tossed at once. Find, correct to three decimal places, the probability of obtaining : (i) exactly 8 heads ; (ii) at least 5 heads ; (iii...

Answer Details

(a) Eight coins, \(n = 8\), \(p = 0.5\) for a head, so each outcome count is \(\binom{8}{r}/256\).

(i) Exactly 8 heads: \(\left(\tfrac12\right)^8 = \dfrac{1}{256} \approx 0.004\).

(ii) At least 5 heads \(= \dfrac{\binom{8}{5}+\binom{8}{6}+\binom{8}{7}+\binom{8}{8}}{256} = \dfrac{56+28+8+1}{256} = \dfrac{93}{256} \approx 0.363\).

(iii) At most 1 head \(= \dfrac{\binom{8}{0}+\binom{8}{1}}{256} = \dfrac{1+8}{256} = \dfrac{9}{256} \approx 0.035\).

(b) The word SHEEP has letters S, H, E, E, P (the E repeats). Choose and arrange 4 letters.

(i) Without restriction, split by how many E's are used:

• Both E's plus 2 of \(\{S,H,P\}\): \(\binom{3}{2}\times\dfrac{4!}{2!} = 3\times12 = 36\).
• At most one E, i.e. the four distinct letters \(S,H,E,P\): \(4! = 24\).

Total \(= 36 + 24 = 60\) arrangements.

(ii) With only one E, the other three must be \(S, H, P\), giving four distinct letters: \(4! = 24\) arrangements.