Example 7.9. A gambler plays a gambling game which costs $1 to participate. He t

Question

Example 7.9. A gambler plays a gambling game which costs $1 to participate. He throus a fair coin repeatedly until he gets a heads. He wins as many dollars as tails he throus before the head is obtained (i) Find the average duration of the game. (ii) Find the expected net gain of each game. Solution: (i) X is the random variable that represents the number of tails before the first head. X is Geometric with p = q E(X) = 1-½ Therefore E(Duration,-E(X + 1 ) = 1 + 1-2 E ( Net Gain ) E ( [# of tails x $1 )1-SI ) = 1-1 = 0 (ii) 50

Explanation / Answer

Game is played until you get the first head.

X represents number of tails before first head

E[x] is the average number of tails

To complete he must get a head after x tails.

Therefore the average duration is taken as E[x+1]=E[X]+1 ie., number of tails + first head.

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