Statistics... my nemesis

[gcalvin]

Cindy,

First, on the specific area of probabilities: sometimes you can help yourself on questions in this area by taking a step back mentally and trying to absorb the "big picture". You mentioned that it's difficult to formulate a rule about adding or multiplying, and trying to key on "and" or "or" in the question. Instead of approaching it that way, try to see the big picture and consider whether a condition is "more likely" or "less likely" than another, and your intuition will guide you correctly.

For example:

You have 100 marbles in a bag. There are 35 blue, 45 red and 20 yellow marbles. Now you pull a marble out at random. What's the probability that it's green? Exactly 0 -- there aren't any green marbles in the bag, right? What's the probability that it's red or yellow or blue? Exactly 1, because those are the three colors in the bag. But we didn't key on "or", we just considered the big picture. What's the probability that it's red? That's .45 -- still clear, right? What's the probability that it's not red? Okay, that's a little tougher, but think of the big picture, take a deep breath, and ask a couple of intermediate questions. Is the answer going to be higher or lower than .45? In other words, is "not red" more likely or less likely than "red"? Maybe by looking at the big picture you can even visualize that "not red" is "a little more than half". In the end, whether you figure out that "not red" is the same as 1 - P(red) = 1 - .45 = .55, or that "not red" is the same as "blue or yellow" = .35 + .20 = .55, you'll come up with the right answer.

Now to make it a little bit harder. You pull out two marbles from the bag. What is the probability that they are both red? Okay, you already know that the probability of the first one being red is .45, and the probability that it's not red is .55, right? So the probability of both being red -- is it higher or lower than .45? In other words, is it more or less likely? Well, it can't be more likely, can it? You had to have the first one be red in order for both to be red. So your answer has to be less than .45, right? (By the way, that may already be enough to choose your correct answer from the choices available.)

Now your next temptation will be to say that the probability of two red marbles is .45 * .45 = .2025. And that can trip you up, because it might even be one of the options on the exam. The trick here is to remember that the probability on that second marble is not quite the same as the first -- it's not 45/100 but 44/99. You understand why, right? So hopefully now you've arrived at 45/100 * 44/99 = .20, the correct answer. A similar approach can help you with lots of other combinations. The important thing is to relax and visualize, rather than try to memorize formulas and rules.

I hope that helped at least a little, and didn't make things even more confusing.

Finally, since you brought up prayer and faith, you can trust that God will bring you through this trial as He has brought you through all the others before now. Try to see things in those terms -- believe that the knowledge is already there from the studying you've done, and trust that it will lead you down the right path. Don't try to force the answers to come, but rather let them come. You know the difference in feeling, attitude and approach, because you have felt it in other subject areas. When you look at a problem, resist the urge to say "oh no! this is one of those probability questions that always give me trouble" and say instead "I have been through enough of these that the answer will come to me if I let it." Then close your eyes and try to visualize as I described above. Maybe even pray over it. Then allow the gifts that God has given you manifest themselves and trust in their strength.

-Gary-