6 Consider the following game of chance A fair coin is tosse

6 Consider the following game of chance. A fair coin is tossed until the first tails appears. You place an initial bet of $k. If the 1st tails appears on the nth toss, you receive a total of $2^n in return for your Initial bet. How large should k be in order for your expected winnings to be zero (note, exported winnings of zero Is sometimes called a \'\'fair\'\' game)? (10 pts)

Solution

Probability of getting tail when a coin is tossed = 0.5

Initial bet is of $ k

If tail appears on the n^th toss you recieve $ 2ⁿ

probability that 1^st tail appears on n^th toss = (probability that head appears on 1^st toss) X (probability that head appears on 2^nd?toss) ... X (probability that head appears on (n-1)^th?toss) X (probability that tail appears on n^th toss) = 0.5^n-1 X 0.5 = 0.5ⁿ

Expected Winning = [ (2ⁿ- k) X (probability that 1^st tail appears on n^th toss)] + [ (-k) X (1 - probability that 1^st tail appears on n^th toss)]

Expected winnings = [ (2ⁿ- k) X (0.5ⁿ)] + [ (-k) X ( 1 - 0.5ⁿ)]

= 2ⁿ X 0.5ⁿ - k X 0.5ⁿ - k + k X 0.5ⁿ

   ⁼   (2 X 0.5)ⁿ - k

= 1 - k ... (I)

Now Sometimes expected winnings of 0 is called a fair game

Suppose Expected Winnings = 0

From (I) we get,

1 - k =0

that is k = 1

So k should be 1 in order to have expected winnings equal to 0.