Thank you Professor Neal, I knew it had something to do with indicators but I couldn't really consolidate it into a solution.
[EMAIL PROTECTED] (Radford Neal) wrote in message news:<[EMAIL PROTECTED]>... > >[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/ . =================================================================
