|
|
|
|
|
|
|
|
|
|
|
|
|
The probability of getting a 6 is 1/6. The expected value is 1*(3/6) + 2*(2/6) + 6*(1/6) = 13/6 or 2 and 1/6. Since the charge is only $2.00 per game, you would expect to win an average of $.16667 each play.
2.
Outcome: Probability: 0 ? 1 .1 2 .2 3 .5
a) What value should replace the question mark if this is
a probability distribution?
b) Is this a discrete or continuous distribution?
Explain.
c) What is the mean of this probability distribution?
d) What is the standard deviation of this probability
distribution?
The missing probability is
.2, needed to make the probabilities add up to 1. It is
discrete because there are only four possible outcomes.
The mean of the distribution is 0+.1 + .4 + 1.5 = 2. The
standard deviation is the square root of (4*.2 + 1*.1 + 0 +
1*.5) = square root of (1.4) = 1.18
3. John has tossed a coin five times and has had this sequence of results: Heads, Heads, Tails, Heads, Tails. He is wondering how likely this result is compared to the sequence: Heads, Heads, Heads, Heads, Heads. What is the right answer?
All sequences are equally likely. The reason that he is likely to get about half heads and half tails is not that the sequences with an even split of heads and tail are more likely but that there are many more sequences of this sort.
4. A game of chance is played by spinning a wheel and paying the amount that comes up. There are four possible outcomes:
x p(x) $1 .50 $2 .30 $5 .15 $10 ??
The missing probability is .05. The expected value is $.5 + $.6 + $.75 + $.5 or $2.35.
5a) Show that P(x) = x/11 - .25 for x = 4, 5, 6, 7 is a probability distribution.
a) 4/11 + 5/11 + 6/11 + 7/ 11
-1 = 22/11-1 = 1; There are no negative probabilities that
this rule assigns.
b) 262/44 = 5.955 (it does not work out to an integer.
c) .4341+1.864 + .0006 + .4216 = 1.043.
d) The probabilities over the sample space do not total 1.
6. Jack has a trick coin that will come up heads two-thirds of the time.
a) P(T) = 1 - P(H) = 1 - 2/3
= 1/3.
b) -1.00*(2/3) + 1.50*(1/3) = -$.5/3 = -$1667. You would
lose on average about 17 cents each time you played this
game.
7. Here is an example of a discrete, uniform probability distribution:
x p(x) 0 .2 1 .2 2 .2 3 .2 4 .2
Find:
What is the expected value of x?
What is the standard deviation of this distribution?
mean = .2(0 + 1 + 2 + 3 + 4)
= .2*10 = 2
st dev = square root(.2(4 + 1 + 0 + 1 + 4)) = square root
(2) = 1.41421....
8. The game show Deal or No Deal features 26 suitcases with various amounts of money. The contestant chooses one than then begins to open the others. At the end of each round, the "Banker" makes an offer to end the game. The game ends either when the player takes the offer, or when he or she has opened all 25 suitcases that he or she did not choose, in which case he or she gets the amount in the chosen suitcase.
($1,000,000 + $100,000 + $1000 + $100)/4 = $1,101,100/4 = $275,275.
9. P(x) = .5 -x/10 is a probability distribution for x = 1, 2, 3, 4.
a) P(4) = .5 - 4/10 = .1;
P(1) = .5 - 1/10 = .4; P(4 and 1) = P(4)*P(1) = .1*.4 =
.04.
Prob(at least one 1) = 1 - (Prob(no 1s) = 1 - (.6)*(.6) = 1
- .36 = .64.