Compute the quantity requested in each problem. Give an exact answer if possible

Question

Compute the quantity requested in each problem. Give an exact answer if possible; otherwise, round to three places after the decimal. If you use a Geometric or Binomial distribution, say so and identify the parameter(s). If you use a CDF, mention how you computed it (chart or command).

1. In a certain game, each player tosses a fair coin until it lands “heads.” Whoever tosses it the most times wins. (The same number of tosses constitutes a “draw” and neither player wins.) Find the probability of winning this game.

Explanation / Answer

P(Player1 wining in 1st attempt) = P(Player1 gets tails in first)*P(Player2 gets heads in first)

= 1/2*1/2 = (1/2)2

P(Player1 wining in 2nd attempt) = P(Player1 gets tails in first)*P(Player2 gets tails in first)*P(Player1 gets tails in second)*P(Player2 gets heads in second)

= 1/2*1/2*1/2*1/2 = (1/2)4

P(Player1 wining in 3rd attempt) = P(Player1 gets tails in first)*P(Player2 gets tails in first)*P(Player1 gets tails in second)*P(Player2 gets tails in second)*P(Player1 gets tails in third)*P(Player2 gets heads in third)

= 1/2*1/2*1/2*1/2*1/2*1/2 = (1/2)6

and so on

Thus, required probability

= (1/2)2+(1/2)4+(1/2)6+(1/2)8+... + (1/2)infinity

This is a Geometric progression with first term(a) = (1/2)2

and common factor = (1/2)4 / (1/2)2 = (1/2)2

Thus,

P(winning) = a/1-r

= (1/2)2 / {1 - (1/2)2}

= 1/4 / (1-1/4)

= 1/4 / (3/4)

= 1/4 * 4/3

= 1/3

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