In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2, ..., 36. (See the photo.) To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots.
b. Determine the probability that the metal ball falls into the slot marked eight. Interpret this probability.

Verified step by step guidance

1. Step 1: Identify the total number of possible outcomes. In roulette, the wheel has 38 slots: numbers 1 through 36, plus 0 and 00. So, the total number of possible outcomes is 38.
2. Step 2: Identify the number of favorable outcomes. Since we want the probability that the ball lands on the slot marked eight, there is exactly 1 favorable outcome (the slot numbered 8).
3. Step 3: Use the formula for probability, which is the ratio of favorable outcomes to total outcomes: \(P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\).
4. Step 4: Substitute the values into the formula: \(P(\text{ball lands on 8}) = \frac{1}{38}\).
5. Step 5: Interpret the probability. This means that if the roulette wheel is spun many times, the ball is expected to land on the slot marked eight approximately 1 out of every 38 spins, assuming the wheel is fair and each slot is equally likely.

Key Concepts

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. In roulette, the sample space consists of all 38 slots (numbers 0, 00, and 1 through 36). Understanding the sample space is essential to calculate the probability of any specific outcome.

Probability of a Single Event

The probability of a single event is the ratio of favorable outcomes to the total number of possible outcomes. For the ball landing on slot eight, the favorable outcome is one (slot 8), and the total outcomes are 38, so the probability is 1/38.

Interpretation of Probability

Interpreting probability involves understanding what the numerical value means in real terms. A probability of 1/38 means that if the roulette wheel is spun many times, the ball is expected to land on slot eight about once every 38 spins, reflecting the likelihood of that event.