Probability in Statistics: 4 Key Points to Better Understanding

Do you have trouble understanding the concept of probability in statistics? Do you ask yourself why you read that section on probability in your statistics book that seems to have no bearing on your research? Don’t despair.

Read the following article and clearly understand this concept that you will find very useful in your research venture.

One topic in the Statistics course students had difficulty understanding is the concept of probability. But is “probability” really a difficult thing to understand?

In reality, it is not that difficult as long as you gain an understanding of how it works when trying to compare differences or correlations between variables.

I explain how probability in statistics in the next sections.

The Coin: Classic Example of Probability in Statistics

I demonstrate the classic example of using a coin to illustrate probability in statistics. Even this simple illustration sometimes eludes the mind of those who have angst whenever they encounter numbers. It is a phenomenon rooted in students’ attitude towards solving math problems.

Now, back to the classic example of probability in statistics using a coin.

Everybody knows that a coin has two sides: the head, which normally has the face of someone on it with the corresponding amount it represents, or the tail, which typically shows the government bank which issued the currency.

Now, if you flick the coin, it will land and settle with one side up; unless you get a weird result that the coin unexpectedly landed on its edge or in-between the head and tail sides! (see Fig. 1).

Fig. 1. Head, in-between, tail (L-R)

This situation, however, could be a possibility as there is a middle ground that will make this possible though very, very remote (what if the government produces a coin thick enough to make this possible if ever you flick a coin?).

I just included this possibility because it so happened I flicked a coin before and it landed next to an object that made it stand on its edge instead of falling on either the head or the tail side. What a weird event with very remote possibility.

That just means that unexpected things could happen given the right circumstances that will make it possible.

An Illustration of Probability

I just have to illustrate this with a picture because some students do not understand what is a head and what is a tail in a coin. So, no excuses for not understanding what we are talking about here.

For our purpose, we’ll just leave the in-between possibility and just concentrate on either the possibility of getting a head or a tail when a coin is flipped and allowed to settle on level ground or on top of your palm.

Since there are only two possibilities here, we can then say that there is a 50-50, 0.5 or 1/2 possibility that the coin will land as head or tail. If we would like to represent this as a symbol in statistics to show this possibility, it is written thus:

p = 0.5

where p is the probability symbol and the value 0.5 is the estimated outcome that the coin will land on either the head or the tail.

Alternatively, this can be stated that there is an equal chance that you will get a head or a tail in a series of tossing a coin and letting it land on level ground.

Therefore, if you toss a coin 10 times, the probability of getting either a head or a tail is 50%, 0.05 or 1/2. That means in 10 tosses, there will probably be 5 heads and 5 tails. If you toss the coin 100 times, you will likely get 50 heads and 50 tails.

Going further, if you have a six-sided dice, then the probability of each side in each throw is 1/6. If you have a cube, then the probability of each side is 1/4.

Application of Probability in Statistics

This background knowledge can help you understand the importance of the p-value in statistical tests.

For example, if you are interested in knowing if a significant difference between two sets of variables exists (say a comparison of the test scores of a group of students who were given remedial classes as opposed to another group that did not undergo remedial classes), and a statistical software was used to analyze the data (presumably a t-test was applied), you just have to look at the p-value to find out if indeed there is a significant difference in achievement between the two groups.

If the p-value is 0.05 or lower than that, then you can safely say that there is sufficient evidence that students who underwent remedial classes performed better (in terms of their test scores) than those who did not undergo remedial classes.

Null and Alternative Hypothesis to Demonstrate Probability

For clarity on the concept of probability in statistics, here are the null and alternative hypotheses that you can formulate for this study:

Null Hypothesis: There is no significant difference between the test scores of students who took remedial classes and students who did not take remedial classes.

Alternative Hypothesis: There is a significant difference between the test scores of students who took remedial classes and students who did not take remedial classes.

The p-value simply means that there is a 5% probability, possibility or chance that students who were given remedial classes perform similarly with those who were not given remedial classes. This probability is quite low, such that you may reject your null hypothesis that there is no difference in test scores of students with or without remedial classes. If you reject the null hypothesis, then you should accept your alternative hypothesis, which is:

There is a significant difference between the test scores of students who took remedial classes and students who did not take remedial classes.

Of what use is this finding then? The results show that, indeed, giving remedial classes can benefit students. As the results of the study showed, it can significantly increase the student’s test scores.

You may then present the results of your study and confidently recommend that remedial classes be given to students to help improve their test scores in whatever subject that may be.

That’s how statistics and probability as an essential component work in research.

©2013 May 15 Patrick Regoniel | Updated 2023 Feb 24