Pretend that the 10 balls are ping-pong balls, that they have been finely ground up, and that the red and white specks have been thoroughly mixed. Forty percent of the specks will be red, and 60% will be white. Suppose that you stir the specks and use a tablespoon to scoop out 10% of the specks. That is, suppose that the tablespoon holds a quantity of specks corresponding to one ball.
(a) What percentage of a red ball is contained in the spoon?
(b) What percentage of the remaining specks in the urn are red?
(c) If the spoonful of specks is replaced, does the percentage of red specks in the urn
change?
(d) Use the results from parts (b) and (c) to explain why the expected number of red
balls as calculated in parts 6 and 7 is the same with and without replacement.
An urn contains four red balls and six white balls. Suppose that two balls are drawn at random from the urn, and let X be the number of red balls drawn. The probability of obtaining two red balls depends on whether the balls are drawn with or without replacement. The purpose of this project is to show that the expected number of red balls drawn is not affected by whether or not the first ball is replaced before the second ball is drawn. The idea for this project was taken from the article "An Unexpected Expected Value," by Stephen Schwartzman, which appeared in the February 1993 issue of The Mathematics Teacher.