RI S2B Binomial Distribution_tutorial (Solns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 _______________________________ Tutorial S2B: Binomial Distribution Page 1 of 11 Tutorial S2B: Binomial Distribution Section A (Discussion Questions) 1 State, with a reason, whether a binomial distribution could be used in each of the following problems. If binomial distribution is an acceptable model, define the random variable clearly and state its parameters. (a) A gambler has a biased coin for which the probability of obtaining a head in any toss is 0.56. Find the probability of getting 6 heads if he tosses the coin 8 times. (b) A fair coin is spun until a head occurs. Find the probability that 8 spins are necessary, including the one on which the head occurs. (c) A jar contains 49 balls numbered 1 to 49. Six of the balls are drawn at random without replacement. Find the probability that four out of the six balls drawn have an even score. Solution: (a) Since there are 8 independent trials (tosses of coins), each toss either yields a head or tail and the probability of getting a head stays constant at 0.56 from toss to toss, binomial distribution is an acceptable model. Let X be the no. of heads obtained, out of 8 tosses.       6 2~ B 8, 0.568P 6 0.56 0.44 0.1676XX      (b) No. For binomial distribution to be used, it should consist of n independent spins but in this case, there are no fixed n number of spins. (c) No. For binomial distribution to be used, the draws should be independent of each other, which is not the case here since the balls are drawn with no replacement. Note: Probability of drawing an even-numbered ball is constant regardless of the draw. Eg P(drawing an even-numbered ball in 1st draw) = 2449 P(drawing an even-numbered ball in 2nd draw) = P(drawing an even-numbered ball in 2nd draw and odd-numbered ball in 1st draw) + P(drawing an even-numbered ball in 2nd draw and even-numbered ball in 1st draw) 25 24 24 23 2449 48 49 48 49     Similarly, P(drawing an even-numbered ball in 3rd draw) =… = P(drawing an even-numbered ball in 6th draw) = 2449

Raffles Institution H2 Mathematics 2025 Year 6 _________________________________________________________________________________________________ ________________________________ Tutorial S2B: Binomial Distribution Page 2 of 11 2 Given ~ B 10, 0.2X , find (i) ; (ii) ; (iii) ; (iv) P(4 8)X  ; (v) P(2 6)X  ; (vi) E X ; (vii) Var X . [ (i) 0.107 (ii) 0.678 (iii) 0.0328 (iv) 0.121 (v) 0.618 (vi) 2 (vii) 1.6] (i) and (ii) involves direct application of graphing calculator. (iii) = 1 P 4X  = 0.0328 (iv) P 4 8 P 8 P 3X X X      0.121 (v) P 2 6 P 5 P 1X X X     