Problems Related To Probability Assignment Help
1.a) In how many ways can 10 cents be distributed among 3 children? (All that matters is how much each child gets, not which coins, that is, cents are considered indistinguishable).
(b) In how many ways if each child is to get at least one cent?
2.a)4 indistinguishable balls are distributed randomly into 3 distinguishable boxes. Let X denote the number of balls that end up in the first box. Find E(X).
b)Suppose that n balls are distributed in k boxes (n ≥ k). Find the probability that there are no empty boxes.
c)Draw randomly without replacement n balls from a bowl containing R red balls and N − R white balls, and let Xj = 1 if the jth draw is a red ball and X j = 0 if the jth draw is a white ball. Show that X1,…, Xn are not independent.
3.An urn contains nR red balls, nB black balls and nW white balls. r balls are chosen at random and with replacement. Find the probability that:
i)All r balls are red;
ii)At least one ball is red;
iii)r1 balls are red, r2 balls are black and r3 balls are white (r1 + r2 + r3 = r);
iv)There are balls of all three colors.
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4.Place five similar balls (each either red or blue) in a bowl at random as follows: A coin is flipped 5 independent times and a red ball is placed in the bowl for each head and a blue ball for each tail. The bowl is then taken and two balls are selected at random without replacement.Given that each of those two balls is red, compute the conditional probability that 5 red blls were placed in the bowl at random.
5.There are 5 red chips and 3 blue chips in a bowl. The red chips are numbered I , 2, 3, 4, 5, respectively, and the blue chips are numbered l , 2, 3, respectivel y. If 2 chips are to be drawn at random and without replacement, find the probability that these chips have either the same number or the same color.
6.Bowl I contains 6 red chips and 4 blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II which was originally empty. One chip is then drawn at random from bowl II. Relative to the hypothesis that this chip is blue. find the conditional probability that 2 red chips and 3 blue chips are transferred from bowl I to bowl II.
7.Bowl I contains 3 red chips and 7.bluechips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl.
(a)Compute the probability that this chip is red.
(b)Relative to the hypothesis that the chip is red, find the conditional probability that it is drawn from bowl II.
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