Sometimes, when you're asked to find the probability of something, it's a good idea to start by splitting the problem into simpler probabilities. For example, the problem may ask, "What is the probability that you will get a video game from Mom AND a box of chocolates from Aunt Sue this Christmas?" Assuming that Mom and Aunt Sue don't consult each other (making the two events independent of each other), you need to know two things:
But once you get the two probabilities, what do you do? Add them? Multiply them? Hmmm...
~
Here are a few general rules to guide you in combining probabilities:
Assume that two events, A and B, are independent (the probability of one doesn't depend on the probability of the other). To find the probability that A AND B will both occur, multiply their individual probabilities.
Example: There is a 1/7 chance that my birthday is on a Sunday. There is a 1/2 chance that it will rain on my birthday. Assuming that these two are independent, the probability that my birthday is on a rainy Sunday (in other words, my birthday will be on a Sunday AND it will rain on my birthday) is 1/14 because
(1/7) * (1/2) = 1/14.
Assume that two events, A and B, are mutually exclusive (they can never happen together). To find the probability that either A OR B will occur, add their individual probabilities.
Example: There is a 2/3 chance that Santa Claus will bring me coal for Christmas. There is a 1/6 chance that Santa Claus will bring me a train set. Since Santa told me that he would only bring me one item, I know that I cannot get both coal and a train set--this means that the events are mutually exclusive and I can use this rule. The probability that I will get coal OR a train set is 5/6 because
(2/3) + (1/6) = 5/6.
What if A and B are not mutually exclusive? Then the probability that either A OR B will happen is a little more complicated. You add the individual probabilities, but you then must subtract the probability that A and B both occur. If you're confused, this Venn diagram might clear things up:
Event A is represented by the blue circle and Event B is represented by the yellow circle. Finding the probability of A OR B means finding the total area covered by those two circles. To obtain this area, you would first add together the areas of the two circles. However, you have now counted the brown area (where they intersect) twice. You only want to count it once, so you must subtract the area of that brown fish-shape.
Example: There is a 1/7 chance that my birthday is on a Sunday. There is a 1/2 chance that it will rain on my birthday. Assuming that these two are independent, the probability that my birthday is on a rainy day OR a Sunday is 4/7 because
(1/7) + (1/2) - [(1/7) * (1/2)]
= 1/7 + 1/2 - 1/14
= 4/7.
To find the probability that an event will NOT happen, simply take the probability that it will happen and subtract that number from 1.
Example: There is a 1/7 chance that my birthday is on a Sunday. There is a 1/2 chance that it will rain on my birthday. Assuming that these two are independent, the probability that my birthday is on a rainy day but NOT a Sunday is 3/7 because
(1 - 1/7) * 1/2
= 6/7 * 1/2
= 3/7.