Bit More About Probability

We base many of our decisions on probabilities. Is it likely to rain tomorrow? What is the probability of a new car breaking down? What are the chances our favor team will win their next game?

We are in search of causation. We want to know if what we see is likely to be due to chance. Or are we seeing a pattern that has meaning. So we begin by calculating the likelihood of events occurring by chance. We calculate probabilities and odd.

Probabilities and odds are related but not identical. They are easy to tell apart because probabilities are stated as decimals and odds are stated as ratios. But the big difference between them is what they compare. Probabilities compare the likelihood of something occurring to the total number of possibilities. Odds compare the likelihood of something occurring to the likelihood of it’s not occurring.

If you roll your ordinary, friendly six-sided die with the numbers 1 through 6 (one on each side), the probability of getting a specific number is .167. This is calculated by taking how many correct answers there are (1), divided by how many total possibilities (6), and expressing it in decimal form (.167). The odds of getting a specific number is how many correct answers (1), against how many incorrect answers. So the odds of rolling a 4 is 1:5…or 5:1 against you.

Let’s try another example. The odds of pulling an King out of a deck of cards is the number of possible correct answers (4), against the number of incorrect answers (48). So the odds are 4:48, which can be reduced to 1:12. The probability of pulling an ace is 4 divided by 52, which equals .077. Probabilities are always decimals, and odds are always ratios.

To calculate the probability of two independent events occurring at the same time, we multiply the probabilities. If the probability of you eating ice cream is .30 (you really like ice cream) and the probability of your getting hit by a car is .50 (you sleep in the middle of the street), the probability that you’ll be eating ice cream when you get hit by a car is .15. Flipping a coin twice (2 independent events) is calculated by multiplying .5 times .5. So the probability of rolling 2 heads in a row is .25. Rolling snake eyes (ones) on a single roll of a pair of dice has a probability of .03 (.167 times .167).

A major question in research is whether or not the data looks like chance. Does the magic drug we created really cure the comon cold or is it a random pattern that just looks like the real thing?

To answer our question, we compare things we think aren’t due to chance to those which we believe are due to chance. We know people vary greatly in performance. We all have strengths and weaknesses. So we assume that people in a control will vary because of chance, not because of what we did to them. But people in different treatment groups should vary because of the experiment, and not just because of chance. Later, we will use this procedure to compare differences between experimental groups to variation within each group. That is, we will compare between-subjects variance to error variance (within-subjects variance).

For the present, we can use the same test (Fischer’s F) to test the significance of a regression. Does the data we collected approximate a straight line? An Analysis of Regression (ANOR) tests whether data is linear. That is, it tests of the fit of the data to a straight line. It, like regression, assumes the two variables being measured are both changing. It works well for testing two continuous variables (like age and height in children) but not so well when one of the variables no longer varies (like age and height in adults).

NOW YOU CHOOSE:
    Day 7: Probability
    Bit More About Probability
    Even More About Probability
    Even More About ANOR
    Calculate ANOR
    Practice Problems
    More Practice Problems
    Word Problems
        Sim1       Sim2        Sim3
    Basic Facts About Probability
    Vocabulary
    Formulas
    Quiz 7
    Summary