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Exam III

(1) What is the expected value, $E(X)$, for the probability distribution in the following table? [Note: the expected value is the same as the mean or average.]

$$\begin{array}{cc} \par Value & Probability \\ -1&.1\\ -1/2 & .4 \\ 0 & .25\\ 1/2 & .2\\ 1 & .05\\ \end{array}$$

(A) $(- 1)^2 (. 1) + (-1/2)^2 (.4) + (1/2)^2 (.2) + (.05)$
(B) $(-1)(.1) + (-1/2)(.4) + (1/2)(.2) + 1(.05)$
(C) $\frac{(-1)+(-1/2)+(0)+(1/2)+1}{5}$
(D) $\frac{1+.4+.25+.2+.05}{5}$

(2) An urn contains three red balls and four white balls. A sample of three balls is selected at random and the number of red balls is observed. Which is a probability distribution for this experiment.


(A) $\begin{array}{cccc} {\rm Outcome} & 1 & 2 & 3 \\ &&&\\ {\rm Probability} & 1/3 & 1/3 & 1/3 \\ \end{array}$ (B) $$\begin{array}{ccccc} {\rm Outcome} & 0&1 & 2 & 3 \\ &&&\\ {\rm Probability} & 1/4 & 1/4& 1/4 &1/4 \\ \end{array}$$

(C) $$\begin{array}{ccccc} {\rm Outcome }& 0&1 & 2 & 3 \\ &&&\\ {\rm Probability }& 4/35& 6/35& 4/35 & 1/35 \\ \end{array}$$ (D) $\begin{array}{ccccc} {\rm Outcome }& 0&1 & 2 & 3 \\ &&&\\ {\rm Probability }& 4/35& 3/35& 3/35 & 1/35 \\ \end{array}$

(E) $\begin{array}{ccccc} {\rm Outcome }& 0&1 & 2 & 3 \\ &&&\\ {\rm Probability }& 4/35& 18/35& 12/35 & 1/35 \\ \end{array}$

( 3) Student A received the following course grades during her fresh- man year: 4,4,4.4,3,3,2,2,2,0. Student B received the following course grades during her freshman year: 4,4,4,4.4,4,3,1,1,1

(A) A has a better GPA and is more consistent.

(B) B has a better GPA and is more consistent.

(C) B has the better GPA but A is more consistent

(D) A has the better GPA but B is more consistent

(E) none of the above.

(4). A bag contains 5 bulbs, two of which are bad. Bulbs are drawn from the bag until a good bulb is removed. The number of draws is observed. The random variable for this experiment has values:

(A) 1,2,3,4,5 (E) 0,1,2,3,4,5 (C) 1,2 (D) 1,2,3 (E) 0,1,2,3

( 5). In American roulette, a better may place a $1 bet an any one of the 38 numbers on the roulette wheel. He wins $35 (plus the return of his bet) if the ball lands an his number, otherwise he loses his bet. What is the expected value of this game?

(A) $1/38 (B) $-37/38 (C) $-1/19 (D) $33.21 (E) $\$ \sqrt{33.21}$

( 6). A single die is tossed four times. What is the probability that exactly two of the tosses show a "one".

(A) $C(4,2) (.5)^2(.5)^2$ (B) $C(4,1)(1/6)^2 (1/6)^2$ (C) $(1/6)^2 (5/6)^2$ (D) $C(4,2)(1/6)^2 (5/6)^2$

(E) $1-C(4,2)(1/6)^2 (5/6)^2$

(7) Suppose that the reliability of a test for hepatitis is specified as follows: of people with hepatitis, $95\%$ have a positive reaction and $5 \%$ have a negative reaction. Of people free of hepatitis, $90\%$ have a negative reaction and $10\%$ have a positive reaction. From a large populations of which $.05 \%$ of the people have hepatitis, a person is selected at random and given the test. If the test is positive, what is the Probability that the person actually has hepatitis?

(A) $\frac{(.95)(.05)}{(.95)(.05)+(.95)(.10)}$

(B) $(.95)(.05)$

(C) $\frac{(.95)(.05)}{(.95)(.10)}$

(D) $(.95)(.10)$

(E) $\frac{(.95)(.0005)}{(.95)(.0005)+(.9995)(.1)}$

(8) What is the variance of the Probability distribution in the following table?

$\begin{array}{cc} Outcome & Prob.\\ 70 & .5 \\ 71 & .2 \\ 72 & .1 \\ 73 & .2 \end{array}$

(A ) $\frac{70^2+71^2+72^2+73^2}{4}$

(B) $70 (.5) +71(.2)+72(.1)+73(.2)$

(C) $70^2 (.5) + 71^2 (.2) + 72^2 (.1) + 73^2(.2)$

( D) $70^2 (.5) + 71^2 (.2) + 72^2 (.1) + 73^2(.2) - 71^2$

(E) $\sqrt{ 70^2(.5) + 71^2 (.2) + 72^2 (.1) + 73^2(.2) - 71^2}$

( 9) Thirty Percent of all cars crossing a toll bridge have a commuter sticker. What is the probability that among ten randomly selected cars waiting to cross the bridge at least two have commuter stickers7

(A) $C(10,2)(.30)^2 (.70)^8$ (B) $1-C(10,2)(.3)^2 (.7)^8$ (C ) $C( 10, 0)(.7)^{10} +C(10,1)(.3)^1(.7)^9$

(D) $1- C(10,0)(.7)^{10}-C(10,1)(.3)^1(.7)^9$ (E) none of the above.

(10) A Person is buying a 10 cent newspaper. He has 2 dimes and 3 pennies in his pocket. The news dealer offers to let him have a paper in exchange for two coins chosen at random from his pocket. Is this a good deal? [Hint: calculate the expected value of this deal]

(A) Yes, it a good deal (B) No, it is a rip off (C) Neither. On the average it will cost the same (D) Can't tell.