Homework

Be sure to state what probability distribution you assume in each problem! You may find the necessary probabilities in tables, compute them manually with a calculator, or use computer software.

PART I

In preparation for a game of "Craps", you were asked the following questions:

a. What is the probability of winning at the first toss of the dice?

b. What is the probability of losing at the first toss?

c. If the first toss is 4, what is the probability of winning on the next toss?

d. What is the probability of throwing a "7 or 11" at least twice in six throws of a standard pair of dice?

e. What is the expected number of 7's & 11's obtained in 6 throws of a pair of dice?

f. What is the expected number of throws of a pair of dice required in order to obtain a 7 or 11?

g. Obtain a pair of dice and throw them six times. How many times did you throw "7 or 11"?

Here are the basic rules of craps. A die is a cube with the faces numbered in such a way that the opposite faces sum to 7, e.g., 3 & 4 are opposite, 1 & 6 are opposite, 2 & 5 are opposite. In the game of "craps", two dice are rolled, and the sum of their uppermost faces is observed.

That is, if the sum has value 2, 3, or 12, the player loses immediately. If the sum is 7 or 11, the player wins. Otherwise (if the sum is 4, 5, 6, 8, 9, or 10) further rolls are required to resolve the game. If, for example, the first sum is 5, then the dice are rolled repeatedly until either a sum of 5 reappears (in which case the player wins) or a sum of 7 appears (in which case the player loses.)

PART II

The foreman of a casting section in a certain factory finds that on the average, 1 in every 9 castings that are made is defective.

a. If the section makes 15 castings a day, what is the probability that 2 of these will be defective?

b. If the section makes 15 castings a day, what is the probability that 3 or more defective castings are made in one day?

c. If the section makes 8 castings a day, what is the probability that 2 or less defective castings are obtained?

PART III

Along highway I-25 in Las Cruces, the probability that each passing car stops to pick up a hitchhiker is p=4%, i.e., an average of one in twenty-five drivers will stop; different drivers, of course, make their decisions to stop or not independently of each other.

a. Each car may be considered as a "trial" in a process, with "success" defined as the car's stopping to pick up the hitchhiker.

b. Given that a hitchhiker has counted 15 cars passing him without stopping, what is the probability that he will be picked up by the 25th car or before?

Suppose that the cars arrive according to a Poisson process, at the average rate of 15 per minute. Then "success" for the hitchhiker occurs at time t provided that both an arrival occurs at t and that car stops to pick him up. Let T1 be the time (in seconds) of the first "success", i.e., the time that he finally gets a ride, when he begins his wait at time t = 0.

c. What is the arrival rate of "successes"?

d. What is the name of the probability distribution of T1?

e. What is the value of E(T1) ? What's the value of Var(T1)?

f. What is the probability that he must wait less than 2 minutes for a ride (P{T1< 2}?

g. What is the probability that he must wait more than 2 minutes for a ride (P{T1> 2}?

h. What is the probability that he must wait exactly 2 minutes for a ride (P{T1= 2}?

Suppose that after 2 minutes (during which 28 cars have passed by) the hitchhiker is still there waiting for a ride.

i. What is the conditional expected value of T1 (expected total waiting time, given that he has already waited 2 minutes)?

Part IV: Adding a profile picture and short biography

Please add a profile picture and short biography to the class Canvas website. Your profile picture has to be acceptable for facial recognition purposes.