>[EMAIL PROTECTED] (Xiao Li) wrote in message >news:<[EMAIL PROTECTED]>... > >> 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?
In article <[EMAIL PROTECTED]>, Ray Koopman <[EMAIL PROTECTED]> wrote: >E(X) = 12*E(Y), where Y is the number of people in any given column >that have someone sitting directly in front of them... Another, simpler, way is to note that E(X) = sum_i E(Y_i), where Y_i is the number of people sitting directly in front of person i (in some arbitrary labelling). Note that Y_i is either 0 or 1, so E(Y_i) = P(Y_i=1). Note also that all the Y_i are the same (nothing distinguishes one person from another a priori). So E(X) = 10 * E(Y_1). If person i sits in the front row, no one will be in front of them. Otherwise, as happens with probability 3/4, the probability that someone will be directly in front of them is just the probability that one of the other 9 people will sit in the chair in front of them. These other 9 people will occupy 9 of the 47 other chairs. The probability that the chair in front is one of the occupied chairs is therefore 9/47. The final answer is therefore E(X) = 10 * (3/4) * (9/47) = 1.43617. This matches the answer obtained by Ray Koopman. ---------------------------------------------------------------------------- Radford M. Neal [EMAIL PROTECTED] Dept. of Statistics and Dept. of Computer Science [EMAIL PROTECTED] University of Toronto http://www.cs.utoronto.ca/~radford ---------------------------------------------------------------------------- . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
