Compute the following binomial probabilities directly from the formula for b (x; n, p):

a. b (3; 8,.35)

b. b (5; 8,.6)

c. P (3<=X<=5) when n=7 and p =.6

d. P (1 < = X) when n=9 and p =.1

B (x; n, p) = nCx (p^x * (1 - p) ^ (n - x))

a.) b (3; 8,.35) = 8C3 (0.35^3 * 0.65^5) = 56 x 0.042875 x 0.116029 = 0.2786

b.) b (5; 8,.6) = 8C5 (0.6^5 * 0.4^3) = 56 x 0.07776 x 0.064 = 0.2787

c.) P (3 < = x < = 5) = P (3) + P (4) + P (5) = 7C3 (0.6^3 * 0.4^4) + 7C4 (0.6^4 * 0.4^3) + 7C5 (0.6^5 * 0.4^2) = (35 x 0.216 x 0.0256) + (35 x 0.1296 x 0.064) + (21 x 0.07776 x 0.16) = 0.193536 + 0.290304 + 0.2612736 = 0.7451