Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In 1959 a world record was set for the longest run on an ungaffed (fair) roulette wheel at the El San Juan Hotel in Puerto Rico. The number 10 appeared six times in a row. What is the probability of the occurrence of this event? (Assume that there are 38 equally likely outcomes consisting of the numbers \(1-36,0\), and 00 .)

Step-by-Step Solution

Answer

The probability of the number 10 appearing six times in a row on an ungaffed roulette wheel is approximately \(\frac{1}{3,010,936,384}\) or \(3.32 \times 10^{-8} \% \).

Step 1: Determine the probability of the single event

The roulette wheel has 38 equally likely outcomes, and we are interested in the appearance of the number 10. The probability of the number 10 occurring can be found by dividing the number of favorable outcomes (1 – only the number 10) by the total number of possible outcomes (38). Thus, the probability of getting 10 is: \(P(10) = \frac{1}{38}\)

Step 2: Determine the probability of the number 10 appearing six times in a row

Since we want to find the probability of the number 10 appearing six times in a row and each spin is independent, we need to multiply the probability of getting the number 10 in a single event six times. The probability formula for independent events is: \(P(A \cap B) = P(A) \times P(B)\) Applying this formula for six events of the number 10 occurring, we have: \(P(10)^6 = (\frac{1}{38})^6\)

Step 3: Calculate the probability

Now let's calculate the probability of the number 10 occurring six times in a row: \(P(10)^6 = (\frac{1}{38})^6 = \frac{1}{3,010,936,384}\)

Step 4: Express the answer as a fraction and percentage

We can express the probability as a fraction: \(P(10)^6 = \frac{1}{3,010,936,384}\) To convert this into a percentage, we can multiply the fraction by 100: \(P(10)^6 \times 100 = \frac{1}{3,010,936,384} \times 100 = 3.32 \times 10^{-8} \% \) So, the probability of the number 10 appearing six times in a row on an ungaffed roulette wheel is approximately \(3.32 \times 10^{-8} \% \).

Key Concepts

Independent Events Probability

Understanding how independent events operate is crucial when calculating the probability of sequential outcomes, such as those in roulette. An independent event is one where the outcome of a previous event does not influence the outcome of a subsequent event. In roulette, each spin of the wheel is independent of the last; the odds of landing on a specific number are the same every time.

When calculating the probability of independent events happening in succession, we multiply the probabilities of each separate event. For example, if the probability of an event A happening is \( P(A) \), and the probability of an event B happening is \( P(B) \), provided that A and B are independent, the joint probability of both events occurring is given by \( P(A \cap B) = P(A) \times P(B) \).

This multiplicative rule allows us to extend the calculation to more than two independent events, such as the occurrence of the roulette number 10 six times in a row, by continually multiplying the probability of the single event.

Probability Calculations

Calculating probabilities is a fundamental skill in understanding chance and risk. Probability is a measure of the likelihood of a particular event occurring, represented as a number between 0 and 1. An event that cannot happen has a probability of 0, while an event that is certain to happen has a probability of 1.

The formula for the probability of an event is given by \( P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \). In the context of roulette, where each outcome should be equally likely, finding the probability of the ball landing on a specific number (like the number 10) involves dividing 1 (the favorable outcome) by 38 (the possible outcomes).

When we have a compound event, like the number 10 appearing multiple times in succession, we need to perform multiple probability calculations, considering the independence of each event. For repeated trials, as we saw in the roulette example, you would raise the probability of the single event to the power equal to the number of trials.

Mathematical Statistics

Mathematical statistics is the branch of mathematics that deals with data collection, analysis, interpretation, and presentation. It provides us with various methods to assess and infer the properties of a population by examining a sample. In the context of games like roulette, it helps us to understand and quantify the variation that arises from randomness.

Statistical analysis also includes computing probabilities, which are the building blocks of statistical conclusions. For example, the historical event of the number 10 appearing six times in succession on a roulette wheel can be interpreted through probability and statistics. The event's improbability tells us that such sequences are rare but possible, given the random nature of independent events in probability.

When presenting probability results, it's often useful to express them in different formats, such as fractions or percentages, for more intuitive understanding. Converting the extremely small probability of the six-fold occurrence of the number 10 from a fraction to a percentage, as in the given solution, makes it clearer just how unlikely the event is. However, even the smallest probabilities are not zero, indicating that in the realm of random events, almost anything can happen given enough trials or time.