Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.
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.
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.
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.
Why do we call it an estimate? Well, unless your sample includes the entire population, there is a chance that you got really unlucky and your sample responses are way off from the way the rest of the population would respond. For example, imagine that you just managed, out of bad luck, to pick all of the students in the school population that have blue as their favorite color. Your probability calculation would be fine for the sample, but would be be way too high for the population! You think you have a .3 probability, but since there are no more blue-preferring students in the school, the real probability is 15/500 = .03. That's a big difference! This is why you have to be careful that you get a good sample, and why this is called estimating the probability of a chance within a given population. It's really only an educated guess!
A probability of chance is a number between 0 and 1 that shows how likely you are to get a certain result in a chance event, where 0 means there is no chance, 1 means that you will always get that result, and any number in between represents your chance of success in that event. In probability calculations, a population is the entire group of objects, persons, etc., to which the probability is being applied, while a sample is that part of the population that you actually test. You calculate probability by dividing the number of successes by the total number of attempts. Your result will be a number between 0 and 1, which can also be expressed as a % if you multiply the number by 100%.
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ISTEP+ Grade 7 - Math: Test Prep & Practice22 chapters | 171 lessons | 8 flashcard sets
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