Question

In a farm with 5746 little piggies:
3192 little piggies went to the market
1988 little piggies had roast beef
A) For this part, assuming going to the market is independent of having roast beef, how many little piggies would you expect have gone to the market but did NOT have roast beef?

B) For this part, if in fact the set of little piggies who went to the market is disjoint from the set of little piggies who had roast beef, what proportion little piggies neither went to the market nor had roast beef?

C) For this part, if in fact 1000 little piggies both went to market and had roast beef, what is the probability that a little piggy had roast beef conditional on that little piggy not having gone to the market?

D) Let's say every little piggy is married to exactly one other little piggy, and all the little piggies who went to the market went with their partner, so exactly 1,596 little piggy couples went to the market. If the price of each little piggy is a random variable, where little piggies (NOT little piggy couples) that went to the market have a mean value of $400 with an SD of $56, and little piggies that did not go to the market have a mean value of $330 with an SD of $42. Let's say that the Big Bad Wolf comes and steals a little piggy COUPLE at random. How much is the farmer expected to lose, and what is the variance of the farmer's expected loss?
Expected loss (write this as a positive value) :_____
Variance:_____

Answer 1

(A)

In case of independence, we can calculate expected frequencies by multiplying row total and column total and dividing by total frequency as follows.

Expected frequency	Went market	Did not go market	Row total
Having roast beef	-----	-----	1988
Not having roast beef	3758*3192/5746 = 2087.632	-----	5746-1988 = 3758
Column total	3192	5746-3192 = 2554	5746

Hence, assuming going to the market is independent of having roast beef expected number of piggies that have gone to the market but did not have roast beef is 2088.

(B)

Observed frequencies are as follows.

Observed frequency	Went market	Did not go market	Row total
Having roast beef	0	1988-0=1988	1988
Not having roast beef	3192-0=3192	2554-1988 = 566 = 3758-3192	5746-1988 = 3758
Column total	3192	5746-3192 = 2554	5746

Hence, required proportion is given by

(C)

Suppose, M and R denote events that went to the market and having roast beef respectively.

Observed frequencies are as follows.

Observed frequency	Went market	Did not go market	Row total
Having roast beef	1000	1988-1000=988	1988
Not having roast beef	3192-1000=2192	2554-988 = 1566 = 3758-2192	5746-1988 = 3758
Column total	3192	5746-3192 = 2554	5746

Required conditional probability is given by

(D)

Number of little piggies that did no go to the market 5746-3192 = 2554.

Suppose, random variables denote values of little piggies that went to the market and random variables denote values of little piggies that did not go to the market.

Farmer's expected loss is given by

Variance of farmer's expected loss is given by