Question

3. A fair coin is flipped eight times and the number of heads is counted. Calculate the probability that the coin will land heads more than 6 times. 4. A coin is flipped 8 times. Calculate the mean, variance and standard deviation

Answer 1

3. 1. Å fair coin is tossed 8 times. so, n=8, p = probability of heads Suppose, x;=so if ith flip results in ooo head. X > xi = Total mo. of heads. From the above situation Xinber (2) :8= I xin Bin (np) where n=8 Þud Because sum of Bernoulli variable is Binomial. Hence the pmf of x is a P(x=x) = ( 6 )* () We want to calculate P(x76) z P(x=7) +P (X=8) (9) (+)?() + C)(3) CDO = 0.03125 + 0.00 391 = 0.03516 An 4. From ② ace get the probability distribution of x where xo - # heads followes Binomial distribution. signidanty now, mean E(X) - Za PCX) I x ( ) ÞX (-6)** ni e E a q in-x71? X! (0-x); pace-pin-x

al Craponna a Ta-12! (-2! det, ka X-1, 50, x=k+! ktl noi me n ! nalok - bac1-6) ! (M - R 1 - *)! n -1-R k sok not nb 2 - Cl- ) - (nol)! bz kso ! (n -1-p)? arponf of Bim (0-100) e np Eb (nal, og pon f is 17 = np.II Because sum of = np. Here, n=8, p = £ so £(x) = 8x1 = 4. (mean) Variance on E (ar) - = {(*) - € (*2 +E(X) -TEC*>]" EE [ *(x-1)] + F(X) – ŏ of [€(*)]" € [*(*-1)] n ! X ( X -1) - (n-x), paccromax o 0 OMS w, for x=0 and x = 1 the above terme comes o.so, the sum starts from a = 2

ni n- a € ( x (x-1)) : 192 (7-2 ) (n -x ) a = kt2 det, kar-2, so, K + 2 B n-2-k n2 ! P C I-E) = 2 in uzo k! CM-2-4)! n-2 md. (n=2)! pre-ben-2-k zmem-) Þr I ko iña 2-k I Bin (M-2,p z nin-ioph [:sum of poof is 1] z o (n-1) br Now, variance onz nen-10 pr+ npan~p~ o miglior ampat op ant or z np (1-P) Hare, n=8,10 26 so , variance a 8x/2x11-12) 2 Standard deviation To Top ciap2 = 1.414,