# Simplified Explanation of Probability in Statistics

Do you have trouble understanding the concept of probability? Do you ask yourself why you have to 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 have a clear understanding of this concept that you will find very useful in your research venture.

One of the topics in the Statistics course that 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 understanding on how it works when trying to compare differences or correlations between variables.

It simply works this way:

The classic example to illustrate probability is demonstrated using a coin. Everybody knows that a coin has two sides: the head, which normally has 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). This, however, could be a possibility as there is a middle ground that will make this possible though very, very remote (what if the government decides to have a coin thick enough to make this possible if ever you flick a coin?). I just included this 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. That just means that unexpected things could happen given the right circumstances that will make it possible.

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 likely be 5 heads and 5 tails. If you toss it 100 times, you will likely get 50 heads and 50 tails.

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

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.

For clarity, 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 provide benefit to students. As the results of the study indicated, 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 work in research.

# The Role of Statistics in Decision Making

At some point in your life, you might be encountering stressful situations that require you to make a choice. How will you be able to make informed and objective decisions? This article describes how statistics can help you out of your misery.

Do you have difficulty in making decisions about personal issues and concerns? Chances are, you are one of those who have decision-making woes especially on personal matters that involve your emotions. How can this be resolved?

The use of statistical tools may help you in this situation. It will help you reduce the uncertainty associated with decision-making that can affect your way of life. It reduces the guesswork related to decision making.

Below is an example of a personal decision-making scenario that demonstrates the role of statistics in decision-making.

### The Role of Statistics in Decision Making

As a practicing statistician for many years, I find the experience of using some tools of statistics like the t-test rather satisfying, especially if I can use it to aid me in decision making. A simple addition of points given for the advantages and disadvantages of a choice may be sufficient in some circumstances, but in some in some instances, more rigorous analysis of statistical data can provide useful information. Statistics can also verify whether the decision made was, after all, a good one.

#### Example Decision-Making Situation Aided by t-test

One concrete, personal experience that demonstrates the role of statistics in decision making happened several years ago. That decision dilemma occurred in 2005. I decided to buy a vehicle to meet a personal and professional need.

I was then very much concerned about the fuel consumption of my second, probably more accurately, third-hand customized owner-type Toyota jeep I bought from a colleague. The jeep guzzles up about 1 liter of gasoline for barely 4 or 5 kilometers of road covered! I thought that this is something that need immediate attention, so I decided to bring the jeep to the automotive repair shop.

I requested the mechanic for a major engine overhaul, where the engine block has to be re-bored to make way for a cylindrical metal sleeve. The metal sleeve narrows the opening where the piston is fired up and down by the series of explosions that occur in the combustion chamber.

However, this engine-related jargon can confound many people unfamiliar with these terms. The whole tune up process aimed to eliminate loose compression of fuel in the engine that leads to a minuscule distance to fuel ratio.

Curiosity struck me, whether my decision to spend for tune-up mattered. Did the number of kilometers covered by the old jeep significantly improve after the tune up?

Armed with a knowledge of the t-test, I sought to find out the answer to this question using the monthly monitoring data I gathered on the total number of kilometers traveled for one gas up. I religiously recorded the number of liters of gasoline in the receipt each time I visit the gasoline station.

I encoded these data in MS Excel, anticipating the t-test computation I will make. I intend to compare the gasoline consumption of my jeepney before and after the tune-up.

Here’s how the simple table where I logged the data on gas consumption looked.

 Gas Fill-up Number No. of km/liter (Before Tune-Up) No. of km/liter (After Tune-Up) 1 2 3 4 5 6 7 8 9 10

The km/liter is computed by simply dividing the total number of kilometers traveled by the number of liters for one gas fill-up.

Example:

100 km/10 liters = 10 km/li

I logged the gasoline consumption of my jeep for several months after the tune up. I then used this data in comparing the jeep’s performance, before and after the tune-up, waiting until I have gathered data for at least ten gas fill ups.

#### Result of the t-test Analysis

So, did I find a significant difference in gasoline consumption before and after the tune-up? The answer is “Yes”. Indeed, the kilometer covered per liter significantly increased almost twice the previous gasoline consumption. It can now run at 8 or 9 kilometers per liter of petrol. That is definitely a better performance than the last 4 kilometers of the road traveled per liter of gasoline. My investment somehow paid off.

This information helped me decide whether I should keep the jeep after having it tuned up in the automotive repair shop. I expected that the jeep should cover more than 8 or 9 kilometers because new vehicles run this distance with the air conditioner on. Moreover, the jeep does not have any air conditioning in it. When the climate is hot, it is hot inside the jeep and when the surrounding environment is cold, well, of course, the jeep is cool too.

#### My Decision After the t-test Analysis

I decided to give the jeep up, sold it and bought a newer, diesel-powered Mitsubishi pickup truck that runs at 11 kilometers per liter of diesel with the air conditioning on. That was five years ago, and my pick up truck still runs like new as I make sure the engine oil is regularly changed to keep the parts within at its best working condition, thus efficiently running.

Maybe I should subject it again to t-test in the coming months to see if I still have to keep it or consider buying a new one. Such is the role of statistics in decision-making.