Question

You are playing a board game with a friend. She owns and has developed the tiles Green Gardens and Silver Sidewalks. If you roll an 8, you will land on Green Gardens and owe her $1500. You will be able to pay, but you will not have money left afterward. If you roll a 9, you will land on the tile King's Taxes and have to pay $75. If you roll a 10, you will land on Silver Sidewalks and owe her $2000, which is more than you can pay—and lose the game. If you roll a 7 or less, you will not have to pay any money to anyone. Your friend offers you insurance. Pay her $500 before you roll and even if you land on Green Gardens or Silver Sidewalks, you will not have to pay any additional money. However, if you roll a 7, you will have to pay her $1000. Use probabilities to find expected values. Then compare the amount of money you should expect to pay out on average, under her insurance and by chance. Is her deal fair?

Answer 1

Answer:

The expected value can be calculated as:

Es = x₁*p₁ + x₂*p2 + .....

where xₙ is the event (the amount of money that you win or lose, and pₙ is the probability for the event)

for example, in a 10 side dice, the probability of obtaining a specific number is 1/10.

we have a d10 i guess (a dice with 10 sides, 1 to 10)

if you roll from 1 to 7, nothing happens.

If you roll an 8, you lose $1500

if you roll a 9, you lose $75

if you roll a 10, you lose $2000 (and the game)

the expected value is,

Es = 1/10*(7*0 - 1*$1500 - 1*$75 - 1*$2000) = -$357

in the second case, you start paying $500.

if you roll a 7, you must pay $1000, if else, nothing happens.

Then the expected value is

Es = -$500 - 1/10*(-1*$1000 + 9*$0) = -$600

So the expected value is smaller in this case, so the deal is not fair