Problem 49 - Algebra and Trigonometry

Question

Roulette An American roulette wheel has 38 slots. Two of the slots are numbered 0 and \(00,\) and the rest are numbered from 1 to \(36 .\) Find the probability that the ball lands in an odd-numbered slot or in a slot with a number higher than \(31 .\)

Step-by-Step Solution

Answer

The probability is \(\frac{21}{38}\).

Step 1: Identify the Total Number of Slots

The American roulette wheel consists of 38 slots, numbered from 0, 00, and 1 through 36. Therefore, the total number of slots is 38.

Step 2: Calculate the Number of Odd-Numbered Slots

The odd-numbered slots in the roulette wheel are: 1, 3, 5, ... up to 35. This sequence forms an arithmetic sequence with the first term as 1 and the common difference of 2, which gives us 18 odd-numbered slots.

Step 3: Calculate the Number of Slots with Number Higher than 31

The slots with numbers higher than 31 are 32, 33, 34, 35, and 36, which totals to 5 slots.

Step 4: Calculate the Overlap Between the Two Events

To avoid double-counting, identify slots that are both odd-numbered and higher than 31, which are 33 and 35. Thus, there are 2 overlapping slots.

Step 5: Use the Addition Rule for Probability

The probability of the ball landing in either an odd-numbered slot or a slot with a number higher than 31 is given by the addition rule \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]where \(A\) is the event "odd-numbered slot" and \(B\) is the event "slot number higher than 31". Using previous steps:\[P(A \cup B) = \frac{18}{38} + \frac{5}{38} - \frac{2}{38} = \frac{21}{38}\]

Step 6: Provide the Final Probability

The probability that the ball lands in either an odd-numbered slot or a slot higher than 31 is \(\frac{21}{38}\).

Key Concepts

American roulette wheel

The American roulette wheel is a popular casino game that features a classic design identifiable by its 38 slots.
What makes the American version stand out is the inclusion of the numbers 0 and 00, along with the numbers 1 through 36.
The wheel is designed to give different outcomes based on the slot where the ball finds rest after spinning. These 38 pockets hold the potential to land players a win, but the presence of both the 0 and 00 slots adds an edge to the house's chances.
In games of chance like roulette, the understanding of probability is key to making educated decisions and managing expectations when playing.

• The total outcome possibilities count up to 38.
• Each slot provides a unique numerical outcome, influencing betting strategies.
• The presence of both 0 and 00 increases the house advantage compared to the European roulette, which only has a single 0.

Knowing the structure of the American roulette wheel helps players understand their chances and informs the application of probability rules discussed in mathematics.

arithmetic sequence

An arithmetic sequence is a series of numbers with a constant difference from one term to the next. This sequence can be seen with the odd numbers on a roulette wheel.
An arithmetic sequence helps calculate the probability of the ball landing in an odd-numbered slot.
For instance, in the American roulette wheel, numbers 1 through 35 are odd and form an arithmetic sequence: 1, 3, 5, ..., 35.
To identify how many odd-numbered slots exist, we can recognize that this sequence has a first term, 1, and a common difference, 2.

• The first term is 1, and from there, you continue adding 2 to reach the next odd number.
• The process stops at the number 35, representing the last odd within the roulette numbers 1 to 36.
• An arithmetic sequence's ability to simplify and visualize slots on the roulette that has common traits is valuable for calculating probabilities effectively.

Understanding sequences is crucial not only for roulette but for sequences in wider mathematics, as it provides a foundational tool for pattern recognition and prediction.

addition rule for probability

Probability rules such as the addition rule are essential in calculating the likelihood of combined events, as seen in the American roulette wheel scenario.
The addition rule for probability applies primarily when trying to find the probability that one out of two or more events occurs.
This rule is particularly handy for workouts involving mutually exclusive or overlapping events.

• The addition rule formula: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
• "\(A\)" in this case represents odd-numbered slots, while "\(B\)" stands for slots numbered higher than 31.
• Due to overlapping possibilities (for instance, slots 33 and 35 in both odd and higher categories), you deduct the overlap to avoid counting it more than once.

Thus, by recognizing overlapping events and utilizing this rule, calculations achieve precision and clarity, essential for determining likelihoods accurately in probabilistic scenarios like gambling or statistical analysis.