Problem 61 - Algebra and Trigonometry

Question

Roulette A roulette wheel has 38 slots. Two slots are numbered 0 and \(00,\) and the rest are numbered 1 to \(36 .\) A player places a bet on a number between 1 and 36 and wins if a ball thrown into the spinning roulette wheel lands in the slot with the same number. Find the probability of winning on two consecutive spins of the roulette wheel.

Step-by-Step Solution

Answer

The probability of winning twice in a row is \( \frac{1}{1444} \).

Step 1: Determine the probability of winning once

To calculate the probability of winning a single bet, consider that there is only one winning number out of the total 38 slots on the roulette wheel. Thus, the probability of winning on one spin is \( P(S_1) = \frac{1}{38} \).

Step 2: Calculate the probability of winning twice in a row

The probability of winning on two consecutive spins is the product of the probabilities of winning on each individual spin. Since the outcomes are independent, the probability of winning twice in a row is \( P(S_1) \times P(S_2) = \left(\frac{1}{38}\right) \times \left(\frac{1}{38}\right) = \frac{1}{1444} \).

Key Concepts

Independent Events

In probability theory, independent events are two or more events where the occurrence of one does not affect the probability of the other. This is a crucial concept when calculating probabilities for sequences of events.
For example, in roulette, each spin of the wheel is independent of the previous spin. The outcome of the first spin does not influence the outcome of the second spin.
Therefore, when you calculate the probability of winning on two consecutive spins, you multiply the probabilities of the outcomes.

• This is because each spin is independent.
• The rule of multiplication for independent events states that the joint probability of two independent events both occurring is the product of their individual probabilities.

Understanding independence helps in correctly evaluating probabilities across multiple events.

Roulette

Roulette is a popular casino game that often baffles new players with its rules and probability. The wheel consists of 38 slots in American roulette, consisting of numbers 1 through 36, plus '0' and '00.'
Players place bets on where they think a ball will land, either on a single number, groups of numbers, or colors. The design of the wheel ensures randomness, meaning each spin is independent of the others.
Here are some key points about roulette:

• Each number has an equal probability of being the outcome on a spin.
• The house edge comes from the presence of '0' and '00,' which add two more slots than the 36 numbered slots.

Playing strategically involves understanding these fundamentals and recognizing that while you can calculate odds, the outcome remains unpredictable.

Probability Calculation

Probability quantifies the likelihood of an event occurring. It's calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
In a scenario like roulette, you calculate the probability of winning once by noting that there’s only one winning number out of 38 slots:
\[ P(S) = \frac{1}{38} \]
This formula means there's a 1 in 38 chance to win on any given spin.

• When calculating the probability of winning twice in a row, an independent event means the individual spins don't affect each other's outcomes.
• This leads us to multiply the individual probabilities:

\[ P(S_1 \text{ and } S_2) = \frac{1}{38} \times \frac{1}{38} = \frac{1}{1444} \]
The multiplication method provides a simple route to understanding more complex probability calculations.