A game is played using one dice. If the dice is rolled and shows 1, the player wins $1; if it shows 2, the player wins $2; if it shows a 3, the player wins $3. If the die shows 4,5,or 6 the player wins nothing.
a.) If there is a charge of $1.25 to play the game, what is the game's expected "gain or loss" for a player?
b.) Suppose a player decided to play this game 100 times, how much would the player expect to gain or lose?
mean win = (1+2+3+0+0+0) = 6/6 = $1
so lose .25 per toss
so lose $25 for 100 games
Two ordinary dice are tossed 360 times.
How many double 4s are there likely to be?
a.) To find the game's expected "gain or loss" for a player, we need to calculate the expected value.
The probability of rolling a 1 is 1/6, and the player wins $1.
The probability of rolling a 2 is 1/6, and the player wins $2.
The probability of rolling a 3 is 1/6, and the player wins $3.
The probability of rolling a 4, 5, or 6 is 3/6, and the player wins nothing.
Using these probabilities and winnings, we can calculate the expected value as follows:
Expected value = (Probability of getting a 1 * Winning from a 1) + (Probability of getting a 2 * Winning from a 2) + (Probability of getting a 3 * Winning from a 3) + (Probability of getting a 4, 5, or 6 * Winning from a 4, 5, or 6)
Expected value = (1/6 * $1) + (1/6 * $2) + (1/6 * $3) + (3/6 * $0)
Expected value = $1/6 + $2/6 + $3/6 + $0
Expected value = $6/6
Expected value = $1
Therefore, the game's expected "gain or loss" for a player is $1.
b.) If a player plays the game 100 times, the expected gain or loss can be calculated by multiplying the expected value by the number of times the game is played:
Expected gain or loss for 100 games = Expected value * Number of games
Expected gain or loss for 100 games = $1 * 100
Expected gain or loss for 100 games = $100
Therefore, if a player plays the game 100 times, they can expect to gain or lose $100.
To find the expected "gain or loss" for a player, we need to calculate the expected value of the game.
a.) To calculate the expected value (EV) for a single play, we multiply the value of each outcome by its probability and sum them up.
Let's calculate the EV for a single play:
- Winning $1 has a probability of 1/6 since there is only one 1 on the dice, and the total number of outcomes is 6.
- Winning $2 has a probability of 1/6 as well.
- Winning $3 also has a probability of 1/6.
- Winning nothing has a probability of 3/6 since there are three numbers on the dice (4, 5, and 6) that result in no winnings.
EV = (1/6 * $1) + (1/6 * $2) + (1/6 * $3) + (3/6 * $0)
= ($1/6) + ($2/6) + ($3/6) + ($0)
= ($1 + $2 + $3) / 6
= $6 / 6
= $1
Therefore, the expected gain or loss for a player is $1 - $1.25 = -$0.25.
b.) If the player decides to play the game 100 times, we can multiply the EV of a single play by 100 to find the expected gain or loss for playing 100 times.
Expected gain or loss for 100 plays = 100 x EV
= 100 x ($1 - $1.25)
= 100 x (-$0.25)
= -$25
Therefore, if a player decides to play the game 100 times, they can expect to lose $25.