Suppose that a person agrees to pay you 10 if you throw at...

Suppose that a person agrees to pay you $$\$ 10$$ if you throw at least one six in four tosses of a fair die. How much would you have to pay him for this opportunity in order to make it a "fair" gamble?

Key Concepts

Expected Value

Expected value is a fundamental concept in probability that represents the average outcome of a random process when it is repeated many times. It is computed by multiplying each outcome by its probability and then summing all these products, which gives a measure of the central tendency or the 'long-run' average of the experiment.

Fair Game

A fair game is a scenario in which neither party has an inherent advantage, meaning the expected gains and losses balance out to zero. In a fair gamble, the entry cost or price to play is set exactly equal to the expected value of the winnings, ensuring that no participant has a statistical edge over the other in the long run.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability that it does not occur. This is particularly useful when calculating the chances of an event happening at least once by determining the probability that it does not happen at all and then subtracting that value from one.

Independent Events

Independent events are events whose outcomes do not affect each other. In probability, when dealing with multiple trials or experiments such as coin tosses or die rolls, independence means that the result of one trial has no influence on the outcomes of the other trials, allowing for simplified calculations of overall probabilities.

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