Calculating the Probability of Chance

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Suppose you'd like to meet a five-foot-tall basketball player in your math class. What are your chances? Here's how to estimate a probability, using a sampling technique and appropriate calculations.

What is Probability of Chance?

Seemingly random things happen around us, all the time. You're randomly drawing objects from a large bag that has three different colors, hoping for a red one, or perhaps you're asking students at your school about their favorite color, hoping the answer will be 'blue'. In this lesson, we will discuss how to estimate the probabilities of a chance event taking place.

A probability of chance is a number between 0 and 1 that expresses the likelihood that a certain chance event will take place. For example, if you have an evenly-weighted coin (neither side is heavier), then flipping the coin is just as likely to land on one side as it is the other. The probability of the coin landing on 'heads' is .5, or one half, which means that half the time you'll get 'heads', and the other half you'll get 'tails'. You can convert the probability to a percentage by multiplying by 100%, which will mean you have a .5 x 100% = 50% chance of 'heads' and a 50% chance of 'tails'. Notice that a probability of 0 means that the event will never happen, and a probability of 1 means that the event is certain - it will happen every time.

Taking a Sample

What if you don't know the probability for a certain chance event? One way to find out is to take a sample from the population of interest. Population is the entire group of things, people, etc., that are of interest to you. It could be the students in your class, or the candy in your 'Trick or Treat' bag, but it's the big picture, which controls all of the possible results. If the population is too large to test completely, you can take a sample. A sample is a portion of the population that will be used to estimate probability for the entire population. It might be ten of the kids from your class, or 15 of the pieces of candy from your 'Trick or Treat' bag. The sample is usually at least one item and fewer than all of the possible items, and the closer the sample size is to the size of the population, the more accurate your probability estimate will be.

For example, say you'd like to know how many students at your school have blue as their favorite color. You could ask everyone, but that would take a long time, so you decide to just randomly ask students and add up the results. If your school has 500 students (the population) and you ask 50 of those students (the sample) what their favorite color is, you can estimate the probability of randomly selecting a student whose favorite color is blue from the school.

Calculating Probability from a Sample

Simple probability is found by dividing the number of successes by the number of possible outcomes. For example, imagine that in your ''favorite color'' experiment you found that 15 of the students said that blue was their favorite color, out of the 50 students you asked. That's 15 actual successes out of 50 possible successes. Dividing the 15 actual successes by the 50 possible, you find that there is a 15/50 = .3 probability of randomly finding a student in that sample whose favorite color is blue.

Now, how does that affect the entire school population? If your sample is a good representation of the students at the school, then you can apply your sample results to the entire population. Instead of surveying all the students, you just estimate the probability, using your result from the sample. You estimate there there is a .3 probability of some randomly-selected student at the school having blue as his or her favorite color.

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