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General Probability Strategies
Now we are in a position to talk about general strategies on probability problems. Something I talked about the last video and I’ll emphasize again. More so with other kinds of math, the hardest part of a probability question is often knowing where to begin. So, in this video I will talk a little more about what you might do to master this very difficult skill.
Where to begin a probability question? It may help you to rephrase the problem in your own words. Especially if that involves the words and or or. In other words, if you can rephrase it, and you wind up with and or or statements, that would be a very good suggestion that you’d be able to use the algebraic rules. So in the previous video we talked about listing, formal algebraic probability rules, and counting techniques.
These are the three broad categories of solutions. For listing or counting solutions, all individual case must be equally likely. Sometimes, one has to reframe what quote unquote, counts as an individual case to identify the cases that are truly, equally likely. So, for example, suppose you’re rolling two die. You wanna know what’s the probability of getting a 7.
Well, a not very good way to frame this would to be say, well you rolled two die. You can get anything from a 2 to a 12, that’s 11 possibilities, it must, getting a 7 must be one out of those 11. You can’t do it that way because the probabilities of getting different roles are different. There’s a different probability of rolling a 3 or a 4 or a 5, those are different probabilities.
What you really have to do is reframe it so you’re looking at the output on each individual dice. So you have six outputs on one die, six outputs on the other dye, there’s actually 36 all together, and then looking at which one of those 36 result in a sum of 7. That’s the correct way to do it, once you’re looking at the, the probabilities that are truly equally likely.
For solutions involving formal algebraic rules it probably will help to identify the conditions as letters. So in other words, if it’s not given to you in algebraic form, write it in algebraic form. That will help you. It may save time to use the compliment rule.
And in particular, if they’re ask you something about the probability of something not happening, in other words if the word not is displayed prominently in the formulation of what the probability is, that is often an excellent clue that it just may be that the complement rule will help you. So always something to keep in mind, that’s one clue that will help you with the complement rule.
For situations involving the idea of or, it will be very important to determine whether events are mutually exclusive. Now let’s review this. The crucial question for mutually exclusive: is it possible for the two events to happen at the same time? And if the answer to that question is yes, then the events are not mutually exclusive.
Then we have to use the general OR rule. If the answer to this question is no. If it is absolutely not possible for the two events to happen at the same time, then they truly are mutually exclusive, and we can use the simple OR rule. For situations involving the idea of AND, it will be important to determine whether the events are independent.
Let’s review this. The crucial question here is: when one event happens, does this change or influence the outcome of the other event? Another way to say this is. Knowing how one event turns out does that give us any information at all about how the other event would turn out?
If the answer is yes, then it’s not independent, then we have to use the conditional probabilities. If the answer is no, if one event truly does not influence the other at all, then the events are independent and then we can use the very simple AND rule. All these are general guidelines, but as I said in the last video, what you are trying to build here are right brain pattern matching skills.
And so there’s gonna be no complete list of rules that’s gonna help you with the pattern-matching of the right brain. The right brain is all about non-linear processes. So the most important thing is to study the solutions and explanations of individual problems. Even if you get the question right, always read the explanation of a probability question.
Because you’re looking at, how do they frame the problem? What is their initial approach? What is the way that they set up the problem? That’s the pattern you’re trying to learn.
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