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Three friends, whom we will call A, B, and C, go to lunch regularly The payer at each lunch is selected randomly until someone pays for lunch for the second time On average, how many times will the group go to lunch Let X denote the number of dinners the group enjoys Clearly, X = 2, 3, or 4 We calculate the probabilities of each of these values If X = 2, then we have a choice of any of the three to pay for the rst lunch Then the same person must pay for the second lunch as well The probability of this is 1/3 If X = 3, then we have a choice of any of the three to pay for the rst lunch Then we must choose one of the two who did not pay for the rst lunch and nally, we must choose one of the two previous payers to pay for the third lunch There are then 3 2 2 = 12 ways in which this can be done and since each way has probability 1/27, the probability that X = 3 is 12/27 = 4/9 Finally, if X = 4 then any of the three can pay for the rst lunch; either of the other two must pay for the second lunch and the one remaining must pay for the third lunch The fourth lunch can be paid by any of the three so this gives 3 2 1 3 = 18 ways in which this can be done Since each has probability (1/3)4 , the probability that X = 4 is 18/81 = 2/9 These probabilities add up to 1 as they should The expected number of lunches is then E(X) = 2 3/9+3 4/9+4 2/9 = 26/9 This is a bit under three, so they might as well go to lunch three times and forget the random choices except that sometimes someone never pays Now what happens as the size of the group increases Does the randomness affect the number of lunches taken.

II Propert i e s p u b l i c i n t Level { get { ret u r n m_leve l j } } p u b l i c bool AllowRec u r s ion { get { ret u r n m_a l l owRe c u r s io n j } } p u b l i c s t r i n g Name { get { ret u r n m_name j } II Met hods public void E n t e r ( ) { TryE nterCore ( As sembly GetCa l l i ngAs sembly ( ) , f a l s e , Timeout I n f i n i t e ) j } p u b l i c void Ente r ( bool p e rmit I n t ra Leve l ) { TryEnterCore ( A s s embly GetC a l l i ngAs sembly ( ) , permit I n t r a Leve l , Timeout I n f i n it e ) j

p u b l i c bool T ry E nt e r ( int m i l l i second sTimeout ) { ret u r n Try E nt e rCore ( As sembly GetC a l l i ngAs sembly ( ) , fa l s e , m i l l i second sTimeout ) j } p u b l i c bool T ry E nt e r ( bool permit I nt r a Leve l , i n t m i l l i secondsT imeout )

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ret u r n T ry E n t e rCore ( As sembly GetCa l l i ngAs sembly ( ) , permit I n t ra Leve l , m i l l i s e c on d s T imeout ) ; } p rivate bool TryEnterCore ( As sembly c a l l e r , boo I p e rmit I n t r a Leve l , i n t m i l l i s e c o n d s T imeout ) { bool t a k e n