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(Math144)[2009](f)midterm~cs_cwmaa^_10457.pdf
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1. How many di.erent linear arrangements are there of the letters A, B, C, D, E, F for which
(a)
A and B are next to each other;

(b)
A is before B and B is before C.


(e.g. {A, F, B, D, E, C}, {D, F, A, B, E, C}, and etc.)
[Solutions: ]
(a)
There are 5!2! = 240 di.erent linear arrangements that A and B are next to each other.

(b)
In total, there are 6! = 720 di.erent linear arrangements of the six di.erent letters. First, there are equal numbers that A is before B and B is before A, i.e. there are 720/2 = 360 di.erent linear arrangements that A is before B. Out of these 360 arrangements, we also have equal numbers that B is before C, C is before B but after A, and C is before both A and B. In other words, there are 360/3 = 120 di.erent linear arrangements that A is before B and B is before C.


2. (a) An ordinary deck of 52 cards is shu.ed. What is the probability that the top four cards have di.erent suits?
(b)
Let P (A)=0.5, P (B)=0.6, and P (A BC )=0.8. Find P (AC B).

(a)
P(the top 4 cards have di.erent suits)


[Solutions: ]
52392613
= 52515049 =0.1055 (to 4 decimal places).
(b) P (AC B)=1 . P (A BC )=1 . [P (A)+ P (BC ) . P (A BC )] = 1 . (0.5+0.4 . 0.8) = 0.9
1
3. You ask your neighbor to water a sickly plant while you are on vacation. Without water it will die with probability 0.8; with water it will die with probability 0.15. You are 90 % certain that your neighbor will remember to water the plant.
(a)
What is the probability that the plant will be alive when you return?

(b)
If it is dead, what is the probability that your neighbor forgot to water it?


[Solutions: ]
Denote the event that the neighbor will remember to water the plant by
W , and that the plant will be dead by D. Thus,

(a) The probability that the plant will be alive when you return is P (DC )=1 . P (D). By the law of total probability, we have
P (D)= P (D|W )P (W )+ P (D|W C )P (W C ) = (0.15)(0.9) + (0.8)(1 . 0.9) = 0.215,
so P (DC )=0.785.
Alternatively, we can .nd P (DC ) directly. Since P (DC |W )=1.P (D|W )=

0.85 and P (DC |W C )=1 . P (D|W C )=0.2,
P (DC )= P (DC |W )P (W )+ P (DC |W C )P (W C ) = (0.85)(0.9) + (0.2)(1 . 0.9) = 0.785.
(b) By Bayes theorem, we have
P (D W C ) P (D|W C )P (W C ) (0.8)(0.1) 16
P (W C |D)= = == .
D 1 . P (DC )1 . 0.785 43
2
4. A standard bottle of the beer advertises that it contains 341 mL of beer. In fact, the machine that pours the beer into the bottle pours a mean amount of 343 mL with a standard deviation of 2mL. The amount of beer poured follows a normal distribution.
(a)
What is the probability that a randomly selected bottle of beer is under.lled?

(b)
If you buy 24 standard bottles of beer, what is the probability that no more than 4 bottles are under.lled?


[Solutions: ]
Let X be the amount of beer poured. So, X N( = 343,2 =22).

(a) The probability that a randomly selected bot