you play a game in which you win $4 if you correctly predict the numbers of spots on a fair 6-sided die and you lose $1 if you don't correctly predict the number of spots. you plan to play 100 times.
how do you set up a box model to represent your net winnings assuming you guess at random?
what is the chance that your net winnings are less than $0?
To set up a box model, you first need to identify the possible outcomes and their associated probabilities. In this case, the possible outcomes are guessing the correct number of spots on a fair 6-sided die, ranging from 1 to 6. The probability of guessing each outcome correctly is 1/6 since all sides are equally likely. The probability of not guessing the correct outcome is 5/6 since there are 5 incorrect possibilities out of the total 6.
Now, you can create the box model representation. Let's say that the outcome "guessing correctly" results in a net win of $4, and the outcome "not guessing correctly" results in a net loss of $1. You can represent this as follows:
Guessing correctly:
- Net win: $4
- Probability: 1/6
Not guessing correctly:
- Net loss: $1
- Probability: 5/6
To calculate the chance that your net winnings are less than $0, you need to consider the sum of the net wins and losses across all the games played. Assuming you play 100 times, you can calculate the expected value of your net winnings.
Expected value = (Net win * Probability of guessing correctly) + (Net loss * Probability of not guessing correctly)
Expected value = ($4 * 1/6) + (-$1 * 5/6)
Expected value = $4/6 - $5/6
Expected value = -$1/6
Since the expected value is negative, it means that over the course of 100 games, on average, you would expect to lose money. In other words, the chance that your net winnings are less than $0 is 100%.