g Suppose I will flip a fair coin until it lands Heads for the first time, and you will receive $2 if the game lasts for 1 round, $4 if the game lasts for 2 rounds, $8 if the game lasts for 3 rounds, and in general, $2n if the game lasts for n rounds. How much would you be willing to pay to play this game once

Step-by-step explanation:

The main idea is that I would like to pay less than what I'm expecting to win, so in that way, I get a profit out of playing this game. Let X be the number of tosses until I get a Heads. By definition, this is a geometric random variable with parameter p = 1/2.

Let Y the amount I received for playing. So, we want to calculate the expected value of Y.

We can calculate it as follows

Since X is a geometric random variable, we have that

Then,

So, we expect to have an infinite amount. Given this, we can pay as much as we want to play the game.

step-by-step explanation:

the original amount doesn't ''re just looking for the time it takes for 1/2 of the substance to we have

(1/2) =)     take the ln   of both sides

ln (1/2)   = )   and we can write

ln(1/2) = ) lne     and lne   = 1     so we have

ln(1/2) = -.1092t     divide both sides by -.1092

t = ln(1/2)/-.1092 = about 6.347 days = about 6.3 days (rounded)

answerthe vertex form

step-by-step explanation: