1 From past experience a wheat farmer living in Manitoba Can

1. From past experience, a wheat farmer living in Manitoba, Canada finds that his annual profit (in Canadian dollars) is $80,000 if the summer weather is typical, $50,000 if the weather is unusually dry, and $20,000 if there is a severe storm that destroys much of his crop.  

Weather bureau records indicate that the probability is 0.70 of typical weather, 0.20 of unusually dry weather, and 0.10 of a severe storm.  

Let X be the farmer\'s profit in the coming year. Construct a probability distribution for the farmer\'s profit to answer the following questions.

(a) What is the probability that the farmer\'s profit is $50,000 or less?

(b) How much profit should the farmer expect to make in the coming year?

2. The state of Georgia has several statewide lottery options. One of the simpler ones is a \"Pick 3\" game in which you pick one of the 1000 three-digit numbers between 000 and 999. The lottery selects a three-digit number at random. With a bet of $1, you win $370 if your number is selected and nothing ($0) otherwise.

(a) With a single $1 bet, what is the probability that you win $370?

(b) Let X denote your winnings for a $1 bet, so x = $0 or x = $370. Construct the probability distribution for X.

(c) The mean of the distribution equals $_________ corresponding to an expected winnings of ___________ cents for the dollar paid to play.

(d) If you play \"PICK 3\" 100 times, how much should you expect to lose?

Solution

The pdf for farmer profit is given below

a) P(X<=50000) = 0.2+0.1 =0.3

b) E(X) = sum of xp(x) = 68000

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a) Prob for winning 370 = 1/1000 = 0.001

pdf

x 0 0.999

370 0.001

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c) Mean = 0(0.999)+370(0.001)

= 0.37

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d) n =100, p of losing = 0.999

Expected value = np (Binomial distribution)

= 99.9

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