Question

You are offered the following gamble based on coin flips. If the first heads occurs on the first flip, you get $2. If the first heads occurs on the second flip, you get $4, and so on, so that if the first heads is on the Nth flip, you get $2N. The game continues until there is a heads. Which of the following best represents the expected value of this gamble in dollars? e 0 π [infinity] When offered, most people say they would pay only less than $10 to play this game.

Answer 1

Answer:

infinity

Step-by-step explanation:

a) the expected value of this gamble in dollars is Infinity

i.e

expected value = $\frac{1}{2}*2 + \frac{1}{4}*4 + \frac{1}{8}*8 + \frac{1}{16}*16 + ... + \to \infty (infinty)$

= $1+1+1+1+1 + ... = \infty$

b)

When offered, most people say they would pay only less than $10 to play this game.

What are two reasons why people are willing to pay so much less than the expected value?

These people are ready to pay less than $10 to play this game due to the fact that people usually overlook the unlikely event when making decisions. In a bid to that logic, they gamble in order to double their amount of money and the probability that heads may never come is ignored by these people and they may hope for a likely event i.e a head every time they play the game.

Also, the expected value is so humongous that if and only if that the first head appears after a long series of tails which is  very less certain to occur, because mostly people would think that on an average the length of a series of tails ( or heads) is somewhat near 10 or so, but definitely infinity.