Problem 1: A national survey that asked adult men and women (not counting the young of less than 20 years of age) revealed that 37% of middle-aged Canadians, more than any other group, believed in Santa Claus. Also, 32% of older Canadians said they believed in Santa Claus, while only 22% of younger adults said they still put faith in Father Christmas. According to the 2011 Canadian Population Census, The adult population of 20+ years of age is split into the three groups (younger, middle-aged and older) as follows: 26% young, 54% middle-aged, and 20% older. (16 points)
a) Construct a table or a tree diagram of joint probabilities.
b) What is the probability that a randomly chosen Canadian adult believes in Santa Claus?
c) If a Canadian adult believes in Santa Claus, what is the probability that this person is of middle age?
d) Are the events Don't Believe and Old mutually exclusive?
e) Are the two events "Don't Believe" and "Young" collectively exhaustive?
Problem 2: A recent survey examined the working arrangements of married couples. It was found that in 30% of all households the man does not work, while the corresponding proportion of households where the woman does not work was twice as much. It was also found that in 60% of the households where the man does not work the woman does not work either.
a) Construct a table or a tree diagram of joint probabilities.
b) What is the probability that in a randomly chosen household both the man and woman work?
c) What is the probability that a randomly chosen household has either person (or both) working?
d) What is the probability that in at least one of two randomly chosen households both the man and woman work?
e) Are working arrangements of the two genders in married households statistically independent?
Problem 3: Your uncle has asked you to analyze his stock portfolio, which contains 6 shares of stock A and 8 shares of stock B. The joint probability distribution of the stock prices is shown below:
|
Price of Stock A - X Variable |
|
|
40 |
50 |
60 |
70 |
| Stock B Y Variable |
45 |
0 |
0.05 |
0.1 |
0.1 |
| 50 |
0.05 |
0.05 |
0.05 |
0.1 |
| 55 |
0.05 |
0.05 |
0.1 |
0.05 |
| 60 |
0.1 |
0.1 |
0.05 |
0 |
a. Find the probability that the price of stock A will be $40 while the price of stock B is $55.
b. Find the probability that either of the two stock prices will be at their maximum price level.
c. Find the average price and variance of stock B.
d. Find the average price and variance of stock A.
e. Can you tell which of the two stocks offers the greatest potential for capital gains?
f. Find the average value of the entire portfolio.
g. Find the covariance and correlation coefficient of the prices of the two stocks.
h. Find the standard deviation of the value of the entire portfolio.
Problem 4: Suppose you invest a fixed sum of money in each of five Internet business ventures. Assume you know the following: (a) 70% of such ventures are successful; (b) the outcomes of the ventures are independent of one another; and (c) the probability distribution for the number, x, of successful ventures out of five is as shown below.
|
x
|
0
|
1
|
2
|
3
|
4
|
5
|
|
P(x)
|
0.002
|
0.029
|
0.132
|
0.309
|
0.360
|
0.168
|
a. Find μ= E(x) and interpret the result.
b. Find and interpret the result.
c. Calculate the interval μ ± 2σ.
d. Graph p(x).
e. Calculate skewness and confirm your interpretation by reference to the graph.
Problem 5: In the game of blackjack as played in casinos, the dealer has the advantage as most of the players do not play very well. As a result, the probability that the average player wins a hand is about 40%. An average player decides to play 10 hands.
f. What is the probability that he will win in all hands?
g. What is the probability that he will win at least 5 hands?
h. What is the probability that he will win at least 1 hand?
i. Assuming that this player has already lost the first 6 hands, what is the probability that he will win all remaining (4) hands?
j. What is the average number of hands that an average player should expect to win in a 10-hand game?
Problem 6: Consider a binomial distribution Bin(n, p) with n = 22.
a. Suppose p = .06. What are the mean and variance of this distribution? For p = .04? for p = .08?
b. In Excel, do the computations necessary to complete the following table.
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x
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P = .04
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P = .06
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P = .08
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Prob. Function
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Cum.
Dist.
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Prob. Function
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Cum.
Dist.
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Prob. Function
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Cum.
Dist.
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0
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1
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2
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3
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4
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5
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6
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7
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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21
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22
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