Question

Use the normal distribution to approximate the desired probability Find the probability that in 212 tosses of a single 15-sided die, we will get fewer than 11 threes. Round your answer to 4 places after the decimal point. Points possible: 2 This is attempt 1 of 2. Submit

Answer 1

The given die is a 15 sided die. Therefore, a P[any value occur in a single toss] = Theutor, P[ twee ocurs in a single toss ] = 3/15. now there are 212 tosses, ljoe Let to were XL,X2... X 212 x = So if be 212 : random variable ith toss has a three ocurs Turefore, xi = 20. op 3/15 бои, 212 YE Š xi : gives after the total no. of 212 tosses of 121 B three ocurs ig sided die a

Novo, 212 12 central limit theorem: since xin Bernouli (b= 3/15) in 1 (1) 212 .. y - B xi ~ Binomial ( 212, þ= %15). Now, owe objective is to find PIY E51]=? {Xm} be a sequence of iid random variable, with E(xi)-H and E(xi?) co . Po, defing on = 5xi now since Y=. Exi - sum of They Sn-E (sh) & N(0,1) 212 iia random variable. Juare (sm) there fore Y-E(r) & N (0,3) 4. This is known 212 121 121 as tworem, Central limit Juar (Y) - uring central limit , D theorem,

Talso, if x~ Bin (n. b) Then E(x) = np. V(X) nop. (lb) Now, ) implies, I-E (Y) N61) Tvar (5) E(Y)= (212 xis) = 14.1833 V(Y) = 212 x 1 x 11-os) = 13.19111 Nowo, Plys 101 = P[ Y-E(Y) E 31 - E(Y) 1 1 (.) is the Juara' Juarry) - CDF of N(013) E91 11 - 84) je $(x)= P(een) i T r since Y-E(4) Vuose (y) , Where AN N10A) Svarty) = $(-0.2800 +534) is follows Nos))

Signature Therefore, P[Y<1] = $(-0.23075). 5 $170.2375) - 10.4061) from Normal distribution talebe. Therefore, answer to 2061 , correct upto 4. decimal