Question

1. (a) The chance of winning a lottery game is 1 in approximately 25 million. Suppose you buy a $1 lottery ticket in anticipation of winning the $75 million grand prize. Calculate your expected net winnings for this single ticket and interpret the result, as indicated below:
µ = E(x) =

Your average LOSSES / GAINS (<—circle the correct all-caps word) would be −−−−−−−−−−−−− (<—fill in the blank) per game.

(b) Now Repeat part (a), but assume a (more realistic) grand prize of $2 million.

(c) Now, repeat part (a), but suppose the chance of winning is 1 in w million, the price of the lottery ticket is y dollars, and that the grand prize is z million dollars. Show that E(x) = z w −y.

(d) Many primary care doctors feel overworked and burdened by potential lawsuits. In fact, a group of researchers reported that 82% of all general practice physicians do not recommend medicine as a career. Let x represent the number of sampled general practice physicians who do not recommend medicine as a career. Why is x approximately a binomial random variable? Use the researchers report to estimate p for the binomial random variable. Consider a random sample of 15 general practice physicians. Use the estimate for p you just found to find the mean and standard deviation of x, the number who do not recommend medicine as a career. Of those 15, find the probability that at least one general practice physician does not recommend medicine as a career. Provide a rough sketch of this binomial distribution, and discuss its shape (is it symmetric?)
Why binomial?
p =
µ =
σ =
P(x ≥ 1) =

Answer 1

Let's do part c) first

c) Expected income (gain) = (1/w mil.)*(z mil) [expectation of a Bin(n,p) random variable is np]

i.e Expected income = z/w

Expenditure = y

Thus, E(X) = Expected income - expenditure = z/w - y (This should be the expectation!)

Let's solve a) and b) now

a) Here z=75, w=25, y=1. Thus, E(X) = 75/25 - 1 = 3-1 = 2. Hence, average gain is $2 per game.

b) Here, z=2, w=25, y=1. Thus E(X) = 2/25 - 1 = -0.92. Hence, average loss is $0.92 per game.

d) Observe that if we denote a practitionar not reccommending medicine as career as 'success', Note that each unit of the sample  are independent (views of doctors regarding this are assumed independent). So, we are actually counting the number of success in the sample of the given size. This by defn is a Binomial random variable.

From the researcher's report, its obvious that an estimate of p is:

Now, we have a random sample of 15 physicians. Thus

The plot of the distribution is given below (the x coordinate is the set {0,1,...,15} and the y-coordinate is the corresponding probabilities. A bar diagram also be a possible visual presentation):

As is clearly evident, it's not symmetric.