Investigation 7: Ice Cream Prices (due Monday, October 11)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones. Integrate appropriate computer output into your report.

Suppose that the owners of an ice cream store hold a promotional special: customers roll two dice, and the price of a small cone in cents is the larger number followed by the smaller number. [For example, if you roll a 3 and a 5 in either order, the price of your ice cream cone is 53 cents.] Denote this price random variable by R. You will investigate the probability distribution of R first through simulation and then through enumneration.

a) Conduct a Minitab simulation with 10,000 repetitions to approximate the probability distribution of this “price” random variable:

MTB> random 10000 c1 c2;

SUBC> integer 1 6.

MTB> rmax c1 c2 c3

MTB> rmin c1 c2 c4

MTB> name c3 'larger' c4 'smaller'

MTB> let c5=(enter formula for calculating price from c3 and c4)

MTB> tally c5

Record the possible values of the random variable along with your empirical estimates of their associated probabilities.

b) Use your simulation results to approximate the probability that the price exceeds 50 cents. Then use your simulation results to approximate the probability that the price is an odd number.

c) Produce (and turn in) a histogram of the approximate probability distribution:

MTB> hist c5

Comment on the appearance of this (approximate) probability distribution as revealed in the histogram.

d) Determine the average price among these 10,000 simulated prices:

MTB> mean c5

e) To see how the average value fluctuates as the number of repetitions increases, use Minitab to compute the (cumulative) average price after each repetition:

MTB> parsum c5 c6

MTB> set c7

DATA> 1:10000

DATA> end

MTB> let c8=c6/c7

MTB> name c8 'cumavgprice'

MTB> plot c8*c7;

SUBC> connect.

Turn in this graph, and comment on what it reveals about how the average price fluctuates over time.

f) Now determine the exact probability distribution of the price random variable R by analyzing the sample space of 36 equally likely outcomes.

g) Did your simulation do a reasonable job of approximating this exact probability distribution? Explain.

h) Determine the theoretical expected value of the price random variable R. Comment on how closely the simulation approximated this value.

i) Use the exact probability distribution to determine the probability that the price R exceeds 50 cents.

j) Use the exact probability distribution to determine the probability that the price R is an odd number.