After having tried all 7 ice-cream flavors of the UConn Dairy Bar Ice Cream Truc

Question

After having tried all 7 ice-cream flavors of the UConn Dairy Bar Ice Cream Truck, you have created your "best of" list of 3. In an end-of-summer event, sampler cups with 5 randomly chosen different flavors from the truck are being distributed. What is the probability that your sample cup contains at least 2 of your favorite 3 flavors? A pair of fair dice is rolled. Let E be the event that the sum of the dice is 7, F be the event that exactly one of the dice is odd, and G be the event that the first die equals 3. (a) Are E and F independent? (b) Are F and G independent? (c) Are E and G independent? d) Are E, F, and G altogether independent?

Explanation / Answer

Q3.

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 5 * 0.42857
= 2.1429
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 5 * 0.42857 * 0.57143
= 1.2245
III.
standard deviation = sqrt( variance ) = sqrt(1.2245
=1.1066

P( X < 2) = P(X=1) + P(X=0) +
= ( 5 1 ) * 0.42857^1 * ( 1- 0.42857 ) ^4 + ( 5 0 ) * 0.42857^0 * ( 1- 0.42857 ) ^5 +   
= 0.2894
P( X > = 2 ) = 1 - P( X < 2) = 0.7106

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