Problem 19 - Algebra and Trigonometry

Question

An American roulette wheel has 38 slots; two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot.

Step-by-Step Solution

Answer

The probability is \( \frac{9}{19} \).

Step 1: Understand the Total Number of Slots

An American roulette wheel has a total of 38 slots. These include numbers ranging from 0, 00, and 1 to 36.

Step 2: Identify the Number of Favorable Outcomes

Among the numbers 1 to 36, odd numbers are 1, 3, 5, ..., 35. To find how many odd-numbered slots there are, we need to count these odd numbers.

Step 3: Count the Odd-Numbered Slots

Odd numbers from 1 to 36 are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35. There are 18 odd numbers.

Step 4: Calculate the Probability

Probability is calculated by dividing the number of favorable outcomes (odd-numbered slots) by the total number of slots. Therefore, the probability is\[ P(\text{odd number}) = \frac{\text{Number of odd-numbered slots}}{\text{Total slots}} = \frac{18}{38} = \frac{9}{19} \]

Step 5: Simplify the Probability Fraction

The fraction \( \frac{18}{38} \) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2. Thus, we get \( \frac{9}{19} \).

Key Concepts

American roulette

American roulette is a popular casino game that involves a spinning wheel with distinctively numbered slots. The wheel consists of 38 slots in total. These slots include two green numbers - 0 and 00 - which give American roulette its unique characteristic compared to other versions, like the European roulette which only has a single 0. The remaining slots are numbered from 1 to 36 and alternate between red and black colors.
These additional slots slightly increase the house edge, making the game marginally more challenging for players. The excitement of roulette largely stems from predicting where the small ball, thrown by the dealer into the spinning wheel, will land. Understanding the structure of the roulette wheel is essential for grasping the probabilities of various bets.

odd numbers

Odd numbers are whole numbers that are not evenly divisible by 2. On an American roulette wheel, odd-numbered slots are those that end in an odd digit. When considering the sequence of numbers from 1 to 36, the odd numbers include 1, 3, 5, 7, and continue in that pattern until you reach 35.
Identifying these numbers is crucial to certain betting strategies where players may choose to place bets specifically on odd numbers. In our exercise, the odd-numbered slots need to be counted to determine the probability of landing on one during a spin of the wheel. This count comes to 18 odd numbers between 1 and 36.

probability calculation

Probability is a measure of the likelihood of a particular event occurring, calculated by dividing the number of favorable outcomes by the total number of possible outcomes. It is expressed as a fraction, a decimal, or a percentage. In the context of roulette, to find the probability of the ball landing in an odd-numbered slot, we perform a simple probability calculation.
For our case, we identified that there are 18 odd-numbered slots out of a total of 38 slots. Therefore, the probability, denoted by \( P \), of landing on an odd number can be calculated using the formula:

• \[ P(\text{odd number}) = \frac{\text{Number of odd-numbered slots}}{\text{Total slots}} = \frac{18}{38} \]

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, resulting in \( \frac{9}{19} \). This simplification yields a more manageable form for further calculations or interpretations.

favorable outcomes

In probability theory, a favorable outcome is an outcome that achieves the event we are interested in. When calculating the probability of a specific event, identifying favorable outcomes is an important step. In our roulette exercise, the favorable outcomes are represented by the ball landing in one of the 18 odd-numbered slots.
These are each of the slots numbered 1, 3, 5, up to 35. The determination of favorable outcomes involves counting all the possible scenarios where the event occurs, which in this case are the odd-numbered slots. By understanding this, one can not only compute the probability effectively but also apply this understanding to a variety of scenarios to predict outcomes and strategize accordingly.