Question

Roulette is a game of chance that involves spinning a wheel that is divided into 38 equal segments, as shown in the accompanying picture. A metal ball is tossed into the wheel as it is spinning, and the ball eventually lands in one of the 38 segments. Each segment has an associated color. Two segments are green. Half of the other 36 segments are red, and the others are black. When a balanced roulette wheel is spun, the ball is equally likely to land in any one of the 38 segments. a. When a balanced roulette wheel is spun, what is the probability that the ball lands in a red segment? b. In the roulette wheel shown, black and red segments alternate. Suppose instead that all red segments were grouped together and that all black segments were together. Does this increase the probability that the ball will land in a red segment? Explain. c. Suppose that you watch 1000 spins of a roulette wheel and note the color that results from each spin. What would be an indication that the wheel was not balanced?

Short answer

a. The probability of the ball landing in a red segment is \(\frac{18}{38}\)\n b. No, grouping the red segments together will not increase the probability of the ball landing in a red segment.\n c. If considerable disparity is observed in the resulting color segments upon 1000 spins, this could be an indication that the wheel is not balanced.

Step-by-step solution

Compute the probability of the ball landing in a red segment

To calculate the probability of the ball landing in a red segment, divide the total number of red segments by the total number of segments. Since half of 36 segments are red, there are 18 red segments. Therefore, the probability is \(\frac{18}{38}\). Simplify this fraction to present the final answer.

Analyze the effect of arrangement on probability

Probability is determined by the number of favorable outcomes divided by total outcomes. Even if the segments are grouped differently, this does not change the number of red segments or total segments. Thus, the probability remains the same regardless of the arrangement and does not increase even if all red segments were grouped together.

Indications of an unbalanced wheel

If a roulette wheel is balanced, each segment, whether red, black or green, should have an equal chance of being selected, which is \(\frac{1}{38}\), over a large number of spins. In 1000 spins, each color should appear approximately the same number of times, in proportion to the segments they have. Major disparity in these numbers, such as one color appearing significantly more than expected, could indicate that the wheel is not balanced.

Key concepts

Probability Calculations

Understanding the odds of different outcomes in a game like roulette starts with grasping the basics of probability calculations. Probability represents the chance that a particular event will occur, and it's expressed as a ratio of favorable outcomes to total possible outcomes.

For instance, when dealing with the roulette scenario given in the exercise, we identify the event of interest (landing in a red segment) and determine the number of ways this event can occur (18 red segments). Next, we divide these favorable outcomes by the total number of possible outcomes (38 segments in total) to get the probability of landing on a red segment, which is calculated as \( \frac{18}{38} \).

To simplify the fraction and present the final answer in its simplest form, we divide both the numerator and the denominator by their greatest common divisor. Understanding how to do this is crucial for clearly conveying the likelihood of the outcome.

Roulette Wheel Probability

The beauty of a roulette wheel lies in its design for equal probabilities among all its segments, assuming it's balanced. Each spin of the wheel gives the metal ball an equal chance to land in any of the 38 segments.

In our roulette exercise, the probability remains constant at \( \frac{1}{38} \) for any single segment because each spin is an independent event. This means that the previous or following spins do not influence the outcome of the current spin. Therefore, rearranging the red and black segments on the wheel without altering their count doesn't change the ball's probability of landing in a red segment which always remains at \( \frac{18}{38} \) or its simplified form.

It is essential to comprehend this independence of events in roulette as it lays the foundation for understanding more complex probability problems and helps debunk common gambler's fallacies about 'due' outcomes.

Indications of an Unbalanced Wheel

As players, we expect a fair game when we approach the roulette table, which means every segment should have an equal chance of being hit. However, physical imperfections or wear and tear could lead to an unbalanced wheel, influencing the probability of where the ball lands.

Looking at large amounts of data can provide statistical evidence of such biases. In our exercise, analyzing 1000 spins gives a significant sample size to gauge if the colors are appearing roughly in proportion to the number of segments they represent. A major deviation from the expected number, especially persistently, can be a red flag for an unbalanced wheel. For example, if the red segments should appear approximately 18 out of every 38 spins, a lower or higher frequency over the 1000 spins can indicate discrepancies. Recognizing these patterns is crucial not only in gaming but also in understanding real-world systems where irregularities can suggest underlying issues.