1. Suppose the probability distribution (pd) for X = number of jobs held during the past year for students in a statistics class is as follows:
X : 0 1 2 3 4
P(X) : .14 .37 .29 .15 .05
(a) find P(x<=2) answer = 0.51
(b) find P(>=1) answer = 0.86
(c) Calculate the E(x) answer = 1.6
(d) Plot the graph of the pd of X.
2. The Smith family wants to have children. They are working with a financial advisor who informs them that historically, families such as theirs the probability distribution of X = the number of children they might have is given as follows:
X: 0 1 2 3
P(X) .05 .60 .30 .05
(a) Calculate E(x) answer = 1.35
(b) Is E(x) a possible outcome for the number of children the Smith family may have?
(c) What is the meaning of E(x)?
3. Californians play a lottery known as Decco. In Decco a player chooses four cards, one each from the four suits, Spades, Clubs, Diamonds, Hearts from an ordinary deck of playing cards. For example a player might choose "5 of clubs", "king of spades", "8 of hearts" and "jack of diamonds". The state draws a winning card from each of the suits. If a players cards match one or more of the states cards, the the player is a winner. It cost $1.00 to play. The payoff matrix or pd is given below. Assume that X= net gain
| Number of matches | Prize | Net Gain | Probability |
| 4 | $5,000.00 | $4,999.00 | 0.000035 |
| 3 | $50.00 | $49.00 | 0.00168 |
| 2 | $5.00 | $4.00 | 0.0303 |
| 1 | Free Ticket | $0.0 | 0.242 |
| 0 | None | $-1.00 | 0.726 |
(a) Calculate the expected value of the lottery answer = -$0.35
(b) What does this tell us?
(c) The free ticket is really not worth the price! Can you explain this?
4. Consider a game in which you roll a die and receive $1 for each spot or dot that occurs. (a) What is the expected winnings for this game? answer = $3.50 (b) If you paid $4 to play the game, how much would you lose, on average, each time you played the game? answer = $0.50
Binomial random Variables
1.Suppose that the probability that you win a game is 0.2 for each play, and plays of the game are independent of one another. Let X = number of wins in three plays. Note this is the probability distribution with number of trials n=3 and probability of success p =.2
(a) Find P(X=2) answer = 0.096
(b) Find P(X>=1) answer = 0.488
(c) Calculate E(X) answer = 0.6
2.For each of the following binomial random variables, specify n an p.
(a) A fair die is rolled 30 times. X = number of times a 6 is rolled.
(b) A company puts a game card in each box of cereal and 1/100 of them are winners. You buy ten boxes of cereal and X = number of games you win.
(c) You like to play computer solitaire in class. You win about 30% of the time. Let X = number of wins in next 20 games.
3. A human and a computer play a 10 game chess tournament. The probability that the human wins any game is 0.6. Find the probability for each of the following events.
(a) They human wins exactly five games. answer = .201
(b) The computer wins exactly seven games. answer = .042
(c) The human wins five games. answer = .834
4. fair cube is colored so that 4 faces are red and 2 faces are green. The cube is tossed 9 times. If success is defined as a red face being up, find:
(a) the mean number of red faces. answer = 6
(b) the probability of getting exactly 5 red faces. answer = .2048
The Normal Distribution
IQ scores follow a normal distribution with mean 100 and standard deviation 15.
What percent of scores will be
(a) between 100 and 115 answer = 34%
(b) what percent of scores will be between 70 and 130? answer = 95%
(c) find the probability that a person selected at random will have an IQ
between 85 and 115? answer = .68
greater than 130 answer = .025