On a roulette wheel are 36 slots numbered 1 through 36
and 2 slots numbered 0 and 00. You can bet on a single
number. If the ball lands on your number, you receive
35 chips plus the chip you played.
a. What is the probability that the ball will land on 17?
b. What are the odds against the ball landing on 17?
c. If each chip is worth $1, what is the expected value
for a player who plays the number 17 for a long time?
a. 1/38
b. 1 - 1/38 = ?
c. How long?
a. To find the probability of the ball landing on 17, we divide the number of favorable outcomes (1) by the total number of possible outcomes.
Total number of possible outcomes = 36 (numbers) + 2 (0 and 00) = 38
Number of favorable outcomes = 1 (17)
Probability of the ball landing on 17 = Number of favorable outcomes / Total number of possible outcomes = 1 / 38 ≈ 0.0263 or 2.63%
b. The odds against an event are found by dividing the number of unfavorable outcomes by the number of favorable outcomes.
Number of unfavorable outcomes = Total number of possible outcomes - Number of favorable outcomes = 38 - 1 = 37
Odds against the ball landing on 17 = Number of unfavorable outcomes / Number of favorable outcomes = 37 / 1 = 37:1
c. To find the expected value, we need to multiply the probability of winning by the amount won and subtract the probability of losing multiplied by the amount bet.
Amount won = 35 chips + 1 chip = 36 chips
Probability of winning = 1/38
Expected value = (Probability of winning * Amount won) - (Probability of losing * Amount bet)
Expected value = (1/38 * 36) - (37/38 * 1) = (36/38) - (37/38) = -1/38 ≈ -$0.0263 or -$0.03
Therefore, the expected value for a player who plays the number 17 for a long time is approximately -$0.03 per chip bet.
To find the answers to these questions, we need to understand the total number of slots on the roulette wheel and how many of them are number 17. Let's break down each question and explain how to calculate the answers:
a. What is the probability that the ball will land on 17?
To calculate the probability, we divide the number of favorable outcomes (the ball landing on 17) by the total number of possible outcomes. In this case, we have 1 favorable outcome (number 17) and a total of 38 possible outcomes (36 numbered slots + 0 + 00).
Probability of landing on 17 = 1 / 38 ≈ 0.026 or 2.6%
So, the probability of the ball landing on 17 is approximately 2.6%.
b. What are the odds against the ball landing on 17?
Odds against landing on 17 can be calculated by subtracting the probability of landing on 17 from 1 and expressing it as a ratio or fraction.
Odds against landing on 17 = (1 - Probability of landing on 17) / Probability of landing on 17
Odds against landing on 17 = (1 - 1/38) / (1/38) = 37/1
So, the odds against the ball landing on 17 are 37 to 1.
c. If each chip is worth $1, what is the expected value for a player who plays the number 17 for a long time?
Expected value is calculated by multiplying each possible outcome by its probability and summing them up.
Expected value = (Payout for winning * Probability of winning) + (Payout for losing * Probability of losing)
In this case, the payout for winning is 35 chips plus the chip you played (36 chips), and the probability of winning is 1/38. The payout for losing is the chip you played (1 chip), and the probability of losing is 37/38.
Expected value = (36 * 1/38) + (1 * 37/38) = 0.947
So, the expected value for a player who plays the number 17 for a long time is approximately $0.947. This means that on average, the player can expect to lose about $0.053 per chip played in the long run.