the game of dots is played by rolling a fair die and receiving 1$ for each dot showing on the top face of the die. What cost should be set for each roll if the game is to be considered a fair game?

expected value

= (1/6)(1) + (1/6)(2) + (1/6)(3) + (1/6)(4) + (1/6)(5) + (1/6)(6)
=(1/6)(1+2+3+4+5+6) = 21/6
= 3.5

so a fair cost would be $3.50

Expected Value

• Making choices by comparing their expected values is a pervasive method of analysis because of its ease of computation and its power. Let’s say you roll a fair die and you were paid, in dollars, the number that landed face up from a fair die roll, what is the most you would pay to play that game? If you were to pay this amount and play this game repeatedly, over a long period of time, how much money would you expect to gain or lose, on average? What if you paid more to play the game? What if you paid less?

What is the answer of is finish first a fair game

To find the fair cost for each roll in the game of dots, we need to consider the expected value. The expected value represents the average outcome of a random variable (in this case, the amount received for each roll) over a large number of trials.

In this game, we know that the die is fair, meaning each side (dot) has an equal probability of landing face up. As there are six sides on a die, each with one dot, the probability of obtaining any specific number of dots is 1/6.

To find the expected value, we multiply each possible outcome (number of dots) by its corresponding probability and sum them up. In this case, let's calculate it:

E(X) = (1 dot * 1/6 probability) + (2 dots * 1/6 probability) + (3 dots * 1/6 probability) + (4 dots * 1/6 probability) + (5 dots * 1/6 probability) + (6 dots * 1/6 probability)

E(X) = (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6)

E(X) = 21/6

Therefore, the expected value is 3.5.

For the game to be considered fair, the cost should be set such that the expected value is equal to zero. In other words, the amount earned should be balanced by the cost of playing the game.

Since the expected value in this case is 3.5, the cost for each roll should also be set at $3.50 to make the game fair.