A roulette wheel has 38 numbered slots around its perimeter, numbered 1 through 36, zero and double zero. It is also divided by color: the odd numbered slots are red, the evens are black, zero and double zero are green. The roulette master spins the wheel, then drops a small ball onto it. The ball eventually settles into one of the slots.
a) What is the probability of the event red or green?
b) What is the probability of the event red and green?
c) What is the probability of the event red and 21?
The Correct Answer and Explanation is :
Let’s analyze the probabilities for the given events on an American roulette wheel, which consists of 38 slots: numbers 1 through 36, 0, and 00. The color distribution is as follows:
- Red: 18 slots (odd numbers from 1 to 10 and 19 to 28, and even numbers from 11 to 18 and 29 to 36)
- Black: 18 slots (even numbers from 1 to 10 and 19 to 28, and odd numbers from 11 to 18 and 29 to 36)
- Green: 2 slots (0 and 00)
a) Probability of the event “red or green”:
This event includes all red and green slots. Since there are 18 red slots and 2 green slots, the total number of favorable outcomes is 18 (red) + 2 (green) = 20. The probability is calculated as:
P(red or green) = Number of favorable outcomes / Total number of slots
P(red or green) = 20 / 38 ≈ 0.5263
Therefore, the probability of the ball landing on a red or green slot is approximately 52.63%.
b) Probability of the event “red and green”:
This event is impossible because a single slot cannot be both red and green simultaneously. Therefore, the probability is:
P(red and green) = 0
c) Probability of the event “red and 21”:
This event is also impossible because slot 21 is black, not red. Therefore, the probability is:
P(red and 21) = 0
Explanation:
In probability theory, the probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For the American roulette wheel:
- Total number of slots: 38
- Red slots: 18
- Green slots: 2
For event (a), “red or green,” we add the number of red and green slots: 18 (red) + 2 (green) = 20 favorable outcomes. Dividing by the total number of slots gives the probability: 20 / 38 ≈ 0.5263.
For events (b) and (c), the probabilities are zero because the specified conditions are impossible. In event (b), a slot cannot be both red and green, and in event (c), slot 21 is black, not red.
Understanding these probabilities is crucial for players to make informed decisions and manage expectations when playing roulette. The house edge in American roulette is approximately 5.26%, which is higher than in European roulette due to the additional double zero slot.