Question

A game consists of tossing a fair coin until heads turns up for the first time. If the coin lands heads on the 1st toss, you get $2. If the coin lands heads on the 2nd toss, you get 2^2=$4. ... If the coin lands heads on the 9th toss, you get 2^9=$512. Finally, if the coin lands heads on or after the 10th toss, you get $1024. What is the expected value (payout) of this game?

Answer 1

We have to define a sample space first.

A game consists of tossing a fair coin until heads turns up for the first time.

We start tossing a coin and we may get Heads first time, we may get heads second time and so on there are infinite possibilities.

S = { H, TH, TTH, TTTH, TTTTH, ......}

THE PROBABILITY OF GETTING head in first trail = 1/2

The probability of getting head in second trail = (1/2)(1/2) { since coin is tossed two times}

Similarly the probability of getting one head in nth trail = (1/2)n

Given we get  $2 for heads on the 1st toss,, 2nd toss, you get 2^2=$4 and it is constant on and after 10th toss

The expected value =

= ( 1+1+1+1+1+1+1+1+1+1+ 1/2 + 1/4 + 1/8 + 1/16 + ....... + 1/2infinity

= 10 +

we know that = 1

Hence The expected value = $11.