Question

(a) The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win: (i) only the second race; (ii) all three races; (iii) only two of the races?
(b) A cone with perpendicular height 24cm has a volume of 1200cm^3. Find the volume of a cone with the same base radius and height 84cm.

Accepted Answer

(a) Let L = probability the athlete does not win a given single race, and W = probability the athlete wins a given single race (W=1-L), assuming each race is independent and identical.
Since P(win none of the 3 races) = L x L x L = L^3 = 1/4, L = (1/4)^(1/3) = 0.63 (2 d.p.).
So W = 1 - L = 1 - 0.63 = 0.37.

(i) P(win only the second race) = L x W x L = 0.63 x 0.37 x 0.63 = 0.147 (approx).
(ii) P(win all three races) = W x W x W = 0.37^3 = 0.051 (approx).
(iii) P(win only two of the races) = 3C2 x W^2 x L = 3 x 0.37^2 x 0.63 = 0.258 (approx), since there are 3 different ways to choose which two of the three races are won.

(b) Since the cones share the same base radius, volume is directly proportional to height (V = (1/3)*pi*r^2*h):
V2/h2 = V1/h1, so V2 = V1 x (h2/h1) = 1200 x (84/24) = 1200 x 3.5 = 4200cm^3.

NECO Mathematics 2022 Theory — Question 12

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