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ID: 3206664 • Letter: C
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Calculators, notes, and books are not allowed. Put your name on back and front of this sheet. Please stop working when time is up. You may leave terms like (52 5) and e^-2 in your answers. What would be a reasonable guess as to the distribution of the following random variable? (a) The number of chocolate chips in the tenth chocolate chip cookie, (b) The number of chocolate chip cookies received until receiving a cookie with exactly 10 chocolate chips, (c) The number of chocolate chip cookies out of the first 10 cookies that have exactly 10 chocolate chips. Suppose we are dealt 6 cards from a well-shuffled standard deck. What is the probability of (a) 4 of one kind and 2 aces? (b) three pairs? (c) three pairs including a pair of aces (that is, a pair of aces and 2 other pairs)? Suppose X has p. m. f. P{X = k} = {25/26 for k = -1/5 1/26 for k = 5 Compute (a) E[X], (b) Var[X], and (c) Ei[X]. Suppose the probability that our team beats A is 9/10, but the probability that our team beats B is 6/10. The probability that we play A first is 2/3; otherwise, we play B first. (a) What is the probability that we win the first game? (b) Given that we win the first game, what is the probability that we played B in that game? (c) What is the probability that we play A first and lose? Suppose the number of days until our ship comes arrives is X where X has a geometric distribution with mean 2 days. Compute the probability that (a) that it is one day until our ship arrives, (b) that it is one day given that it is at most 3 days, and (c) that it is more than 5 days. (a) According the APS Investigative Report, a class should be flagged with probability 1/740. Suppose we have 1480 classes that are independently flagged, each with probability 1/740. Accurately approximate the probability that exactly one of the 1480 classes is flagged. Your answer may include expressions of the form e^x but your answer should not include expressions of the form (n k) or factorials. (b) Suppose X has mean 0 and variance 1. Can you obtain an upper bound on the P{|X| 5}? (c) Suppose Y 0 and the E[Y^3] = 1. Can you obtain a good upper bound on P{Y 3}?Explanation / Answer
a) Only 4 aces are there , so in 4 of a kind we have to exclude the aces set since we have to select 2 aces anyway.
first we choose 1 denomination out of 12 ( excluding ace so 12 instead of 13)
=12C1
4 of a kind from 4 types
=4C4
2 aces from 4 available = 4C2
Total ways to dealt 6 cards = 52C6
Probability = (12C1 X 4C4 X 4C2 )/ 52C6 = 72 / 20358520
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(b)Now we have to select three pairs
there are 13 denominations , choosing a denomination ( 1 for each pair) = 13 C 3
Now 4 suits are there , choosing three pairs = 4C2 X 4C2 X 4C2
Total ways = 52C6
Therefore, Probability = (13C3 X 4C2 X 4C2 X 4C2)/52C6 = 0.00303440525
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(c)A pair of aces can be selected = 4C2
Two pairs = Denomination now remain 12 = Two pairs denomination can be chosen in 12 C2 ways
There are 4 suits = choosing two pairs = 4C2 X 4C2
Total ways = 52C6
Therefore probabibility = (12C2 X 4C2 X 4C2 X 4C2 )/52C6
= 14256/20358520
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