Question

An unfair coin has probability p = 0.20 of being heads and is flipped 100 times. (a) What is the exact distribution, mean, and variance of X the number of heads that appears in this experiments? (b) What is the exact probability that between 15 and 28 heads appear? Please express your solution as a summation. (c) Using the Central Limit Theorem, what is the approximate probability that between 15 and 28 heads appear? Please simplify your solution. (Do not use continuity correction.) (d) Using continuity correction, refine your approximation in part (c). Do not simplify.

Answer 1

(9) since the coin is being flipped independentin and the probability of sucress (Heads) is constant, the distribution of nooit is binomiol. x = nou of heads that appear in 100 trials, * * Bin ( n = bo, b = 0) Hesse ne no. of trials =100 - b = probability of success = 0.2o. irbs - 0.20 0.80 If to Bin (np), Hen Elx) = nb 4 V (+9= nb carb) So, Elt) no 100 (0 : 20). 20 vex) = np Cirble loo (0.2o) (ango) vet) : 161 ab) The exact probability that between 15 and 28. hoods appear = PC 15 4X428) 28, .-Y = A ( 36(*** (since P(+=x) = (?) * (16) X- .. Y =15 no

If neobso, then ~ N(0,1). z t-nb Inpq = r C.L.T. I P( 15 L x 2 28) a pl 15.€ 7 € DETA) = P(-1.25€ 7 22) = P(Z = 2) - P(Z L-1.25) = $(9) - 5 (-1.25), ſuperre & (3)= plz & 2); Z ~N101)] = 0.9772 -0.1056 = 0.8716 Ans,

(d) If tn Bin (m, p); and no, ho, then 2 = tonb ~NCO,1) nby mechon e consecfactor and p(x=x)& pla-np-0.50 zs mba continuity x-ubt0.50 weare z~Noi). pl 15 LV 698) Pi 15-20-0.50 28. 20+ 0. 50 czu =pl-1. 375 CZ = + 2.125) = $( + 2.125) - 5 (-1, 375) = 0.9832 - 0.08 4 6 0.8986. witth of each rectangle is 1. →x 28+0.50 28 15-6.50 15