## Introduction

In probability theory, the probability of rolling a 4 or a 6 on one toss of a standard six-sided die is. The probability of an event is a measure of the chance that the event will occur.

## Theoretical Probability

Theoretical probability is calculated by taking the number of ways an event can happen and dividing it by the total number of possible outcomes. In this case, there are two ways to roll a 4 or a 6 (rolling a 4 and rolling a 6), and there are six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6). Therefore, the theoretical probability of rolling a 4 or a 6 on one toss of a standard six-sided die is 2/6, or 1/3.

### Basic Terminology

There are three basic concepts in probability that you need to understand before we can move on: random variables, events, and sample spaces.

A random variable is a variable whose value is a numerical outcome of a random phenomenon (like the roll of a die). Random variables can be either discrete or continuous.

-Discrete random variables take on values that are separate and distinct from each other (like the numbers on a die).

-Continuous random variables take on values that are part of a continuous scale (like height or weight).

An event is any set of outcomes of a random phenomenon (like rolling a 6 on a die).

The sample space is the set of all outcomes for a particular experiment (like rolling a die).

### Theoretical Probability Formulas

There are two ways to calculate theoretical probability: relative frequencies and set theory. Relative frequencies are used when you have a large number of trials, and set theory is used when you have a small number of trials.

The relative frequency formula is:

P(A) = number of times event A occurs / total number of trials

For example, if you roll a die 100 times and event A is rolling a 3, then P(A) would be 16.7% (16.7 = 1/6 * 100).

The set theory formula is:

P(A) = number of outcomes in event A / total number of possible outcomes

For example, if you have a deck of cards and event A is drawing an ace, then P(A) would be 4/52 (4 = 1/13 * 4, where 1/13 is the probability of any ace and 4 is the number of aces in a deck).

## Experimental Probability

Many people believe that the probability of rolling a 4 or 6 on a standard six-sided die is 1/3. However, this is not the case. The actual probability of rolling a 4 or 6 is 1/6. This means that if you roll the die six times, you would expect to roll a 4 or 6 once.

### Conducting an Experiment

In order to answer this question, we need to conduct an experiment. We will roll a standard six-sided die 60 times, and record the number of times we get a 4 or a 6.

Our null hypothesis is that the probability of rolling a 4 or a 6 is 1/6. Our alternative hypothesis is that the probability of rolling a 4 or a 6 is not 1/6.

We will use a significance level of 0.05 for our test.

### Calculating Experimental Probability

There are a number of ways to calculate experimental probability. The most common way is to find the number of favorable outcomes over the number of possible outcomes. This can be written as a fraction, or it can be expressed as a percentage.

Another way to calculate experimental probability is by using a table or chart. To do this, you need to find the total number of trials and the number of trials that resulted in the desired outcome. For example, if you flipped a coin ten times and got heads five times, your experimental probability of getting heads on one flip would be 50%.

Remember, experimental probability is based on data from repeated trials. The more trials you have, the more accurate your results will be.

## Comparing Theoretical and Experimental Probability

### Why Theoretical and Experimental Probability Might Differ

There are two types of probability: theoretical and experimental. Theoretical probability is what we expect to happen, based on the laws of probability. Experimental probability is based on what actually happens when we run an experiment.

Theoretical and experimental probability can differ for a number of reasons. First, experiments are subject to chance. That means that even if the theoretical probability of something happening is very low, it’s still possible that it will happen in a given experiment.

Second, experiments are usually run with a limited number of trials. That means that even if the experimental probability of something happening is very close to the theoretical probability, there could still be some difference between the two numbers simply because not enough trials were run to get an accurate estimate of the experimental probability.

Lastly, people are not perfect. We might make mistakes when we’re running experiments or recording data. Those mistakes can cause the experimental probability to be different from the theoretical probability.

## Conclusion

In conclusion, the probability of rolling a 4 or a 6 on one toss of a standard six-sided die is 2/6, or 1/3.