Question

a game is played in which a player rolls one 6-sided die and wins $4 if a number less than 4 is rolled, losses $28 if a 6 is rolled, and wins $5 for all other outcomes. let the random variable x denote the amount a player could win/lose if they play the game once.

Answer 1

The random variable X in this game represents the amount a player could win or lose if they play the game once. The possible outcomes of rolling a 6-sided die are numbers less than 4, a 6, and all other outcomes. Based on the given information, if a number less than 4 is rolled, the player wins $4. If a 6 is rolled, the player loses $28. For all other outcomes, the player wins $5.

Step-by-step explanation:

The random variable X in this game represents the amount a player could win or lose if they play the game once. The possible outcomes of rolling a 6-sided die are numbers less than 4, a 6, and all other outcomes. Based on the given information, if a number less than 4 is rolled, the player wins $4. If a 6 is rolled, the player loses $28. For all other outcomes, the player wins $5.

To find the value of X, we need to calculate the amount a player could win or lose for each outcome and their respective probabilities:

1. If a number less than 4 is rolled, the player wins $4 with a probability of 3/6 (since there are 3 numbers less than 4 out of 6 possible outcomes). Therefore, the player could win $4 * (3/6) = $2.
2. If a 6 is rolled, the player loses $28 with a probability of 1/6 (since there is only 1 outcome of rolling a 6 out of 6 possible outcomes). Therefore, the player could lose $28 * (1/6) = $-4.67.
3. For all other outcomes, the player wins $5 with a probability of 2/6 (since there are 2 outcomes that are neither less than 4 nor 6 out of 6 possible outcomes). Therefore, the player could win $5 * (2/6) = $1.67.

Therefore, the random variable X can take on the values $2, $-4.67, and $1.67, depending on the outcome of rolling the die.