You bet! You roll a die. If it comes up a 6, you win 100 ....

You bet! You roll a die. If it comes up a $6,$ you win $\$ 100$ . If not, you get to roll again. If you get a 6 the second time, you win $\$ 50 .$ If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?

You bet! You roll a die. If it comes up a $6,$ you win $\$ 100$ . If not, you get to roll again. If you get a 6 the second time, you win $\$ 50 .$ If not, you lose.
a) Create a probability model for the amount you win.
b) Find the expected amount you'll win.
c) What would you be willing to pay to play this game?

Key Concepts

Probability Model

A probability model is a mathematical representation of a random experiment that defines the sample space (the set of all possible outcomes), the events of interest, and the probabilities associated with those outcomes. In this context, constructing a probability model involves identifying the possible winnings from the game along with the probabilities for each monetary outcome. This model provides a framework for analyzing random processes in order to predict expected outcomes and variabilities.

Expected Value

Expected value is a key concept in probability that represents the long-term average outcome of a random process after many repetitions. It is computed by summing the products of each outcome's value and its respective probability. In decision-making and game analysis, the expected value helps in assessing whether a game is favorable and can guide decisions such as how much one might be willing to pay to participate.

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values, each associated with a specific probability. In the context of this game, the winnings (such as $100, $50, or a loss) are discrete outcomes, and the associated probabilities for these outcomes form the basis of the random variable that models the game’s payoffs.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that a related event has already occurred. In sequential games or experiments, such as rolling a die twice where the second roll only happens after a certain outcome on the first roll, conditional probability is used to calculate the likelihood of combined events. This concept is crucial for understanding the chained structure of outcomes in the game and properly modeling their probabilities.

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