Statistics for Management: Problems on Sampling and Testing

STATISTICS FOR MANAGEMENT: PROBLEMS ON SAMPLING AND TESTING

1. In a game of chance, each player is given an urn containing one five-rupee coin, two two-rupee coins and three one-rupee coins; note that the urn contains six coins altogether. The player draws three coins at random and without replacement. Let X, Y and Z denote the values of these three coins in rupees. Define

   M = median ( X, Y , Z),   L = min (X, Y, Z),   U = max (X, Y, Z),        S = (L + U)/2

Find (a) P(M = 1) , (b) P(M = 2), (c) P(S = 1), (d) P(S = 2), (e) P(S = 3), (f) P(S < M) and  (g) E(S).

2. An alchemist visited the court of a medieval warlord and said “Your excellency, here is my tribute to you. I have six envelopes. One of these contains a single copper coin, another contains two copper coins, while a third one contains three copper coins. The remaining three envelopes are empty. Kindly pick up any three of these six envelopes at random and without replacement. I shall convert all the coins in the selected envelopes to gold coins dating from the period of King Solomon – you can imagine their value as antiques !”

“But what happens if I end up picking only the three empty envelopes?”, thundered the warlord, “I shall behead you then.”

“Take it easy, your excellency”, calmly replied the alchemist “I am also a sorcerer – in that extreme case, I shall make seven gold coins for you, again dating from King Solomon’s era, simply from the air.”

Assume that all the claims of the alchemist were true and that he kept all his promises (the latter point is natural given the threat about his head!). Let X be the number of gold coins that the warlord eventually ended up with. Obtain (a) P(X = 3), (b) P(X = 4), (c) P(X = 5), (d) P(X = 6), (e) P(X = 7), (f) E(X)  and (g) Var(X).