[EMAIL PROTECTED] (Xiao Li) wrote in message news:<[EMAIL PROTECTED]>... > This problem was on my Probability final a month ago... and to this > day, I still haven't figured it out yet. Here it is: > > A theater has a set of seats consisting of four rows and twelve > columns for a total of 48 seats. 10 people walk into the theater and > each person takes a seat at random. Let X be the number of people > with someone sitting directly in front of them. What is the expected > value of X? > > Note: > > In the following case, Sally is sitting directly in front of Mike: > Empty Seat > Sally > Mike > > In the following case, Fred is NOT sitting directly in front of > George: > Fred > Empty Seat > George
People can sit in the columns in the groupings (4, 4, 2), (4, 3, 3) and so on. If 4 people sit in a column, exactly 3 people have someone sitting in front of them. If 1 person sits in a column, then nobody is in front of that person. For 3 persons in a column there are 2 possibilities, 2 persons with somebody in front and 1 person with somebody in front, and for 2 persons there are also 2 possibilities, namely 1 person with somebody in front and 0 persons with somebody in front. Consider the grouping (3, 3, 3, 1). The number of ways in which this can be achieved is C(10, 3)�C(7,3)�C(4,3)�C(1,1)�4!/(3!1!)=67200. This value has to be found for each of the other groupings and the total found. Let's say that this total is T. The probability of the grouping (3, 3, 3, 1) is 67200/T. There are 8 permutations for the columns with 3 people, each with probability 1/8. The numbers of such persons in these permutations are 6, 5, 5, 4, 5, 4, 4, 3 respectively. For the grouping (3, 3, 3, 1) the contribution to the expected value is (6+5+5+4+5+4+4+3)/8�67200/T. This has to be done for all the other groupings and the total taken to give the expected value. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
