Home An Introduction To Probability & Statistics

# An Introduction To Probability & Statistics

The new academic year is well underway and math students around the country are being introduced to new and exciting mathematical concepts. Some of these concepts are completely new, whereas some of them build upon other lessons that students will have had in previous years. Some of the younger middle school students out there are about to experience one of the more interesting encounters that await them during their education: their introduction to probability and statistics.

Probability and statistics are two key pillars in mathematics, and they are two of the more common concepts that we use in everyday life. For middle school math students, probability and statistics are core branches of mathematics that will help them understand and solve much more complex problems down the line.

For today, let’s start at square one. Let’s take a look at some of the fundamental elements of probability and statistics, such as the measures of central tendency and the probability of events. Even by scratching the surface of these two groups of topics, we will learn an incredible amount about probability and statistics, and we will begin to see how and when we can use these tools in our day-to-day lives.

## Important Statistical Terminology

If you were unfamiliar with probability and statistics, and I asked you to list the measures of central tendency, you might look at me as if I’m mad. Even if you are well-versed in this area of mathematics, you may still scratch your head in wonder. The reason for this is that the measures of central tendency are far more famous for their individual names: mean, mode, median, and range.

Each of these terms can be used to calculate something about a particular set of numbers. With this information, we can begin drawing conclusions about that set of numbers. Let’s take a closer look at each of the terms to understand better.

### Mean

The mean is another name for the average, and it is one of the most popular statistical measures that we use. We measure the mean by adding the values of each number in the set together and then dividing by the total number of values.

### Median

The median is the middle number in a set of numbers. When we have an odd number of values, the median is simply the middle number, whereas if we have an even number of values, we add the two middle values together and divide by 2 – finding the mean of these two values – and this is our median.

### Mode

The mode is another measure of central tendency that is similar to the mean and the median, and that’s why these three are often taught and learned together. The mode is the value that appears most times in the set. Our set can have one mode (unimodal), many modes (multimodal), or it can have zero modes if each value in the set occurs with the same frequency.

### Range

The final measure of tendency we need to know as budding statisticians is the range. This is the difference between the largest and smallest numbers in our set. Calculating the range is simple, just take the smallest number away from the largest number and we have our range.

## Probability of Events

Now that we know the measures of tendency and how to calculate them, we can think in more depth about probability and learn how to calculate the probability of certain events. Probability is the measure of the likelihood of an event occurring. When we are talking about probability, we always express the probability of an event happening as a number between zero and 1. Zero means it will not happen, and 1 means it will definitely happen.

Let’s look at an example of probability involving a random event: rolling a six-sided die. Whenever we throw a fair die, there is a 1/6 chance each time that we will roll a particular number. We got this number because we have 1 desired outcome and we have divided it by all of the possible outcomes. That means the probability of rolling a 2 is 1/6. This 1/6 is our answer, and can also be expressed as 0.16666667, or about 17%.

When calculating the probability of events, we use different calculations to find out the probability of multiple events and mutually exclusive events.

### Probability of Multiple Events

If we want to find out how likely it is that several events will occur, we need to find the probabilities of each event occurring individually and then multiply these two values together. This is sometimes known as the “and” probability, and any time you see the word “and” in probability, your mind should jump to multiplication. On the other hand, when you hear the word “or” you should think about addition.

Let’s turn back to our fair die for an example. If we want to find out the probability of rolling a 4 and a 5, our equation would be 1/6 multiplied by 1/6. This is because each time, we have one desired outcome from 6 possible outcomes. In this case, our answer is 1/36 or 0.027.

### Probability of Mutually Exclusive Events

Mutually exclusive events are events that cannot happen simultaneously. This throws a small spanner in the works of our probability calculations. This is known as the “or” probability, and as we mentioned earlier, we will need to perform some addition this time.

Sticking with our example above, what are the chances that we roll a 4 or a 5 with two rolls of a fair die? In each case, the probability of each event remains the same, 1/6. This time, however, we will add the terms together, to find the probability of these two mutually exclusive events. When we do this, we get 1/6 + 1/6 = 1/3 = 0.33333.

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