In part one, I introduced this set of facts:
Augie is blind. Thanks to his organized systems he is able to live alone. However, one day he finds that his laundry service has forgotten to pair his socks together. Augie has 12 black socks, 8 white socks, 6 blue socks and 4 red socks. Augie puts two random socks on.
Now, let’s ask a probability question.
What is the probability that Augie has put on a matching pair of socks?
This question tests both OR and AND probabilities. Let’s lay it out systematically.
There are four ways Augie could be wearing a matching pair of socks (black, black), (white, white), (blue, blue) or (red,red).
Since any one of these situations satisfies our condition of matching socks we want to add their individual probabilities together.
P(matching black) + P(matching white) + P(matching blue) + P(matching red) = P(matching pair of socks)
Now, we need to find those four probabilities individually. In order to pull a matching pair of black socks. In order to do that the first sock we pull out must be black. The odds of that are 12/30 (number of black socks/number of total socks) (12/30 reduces to 2/5 but we’ll save reducing for the end). But in order to pull a pair of black socks we need to pull a black sock first AND pull a black sock second. In order to pull the second black sock we’ll have to pull one of the remaining 11 black socks out of the remaining 29. Since we need both events to occur in order to get a black pair, we’ll multiply the individual probabilities to get the probability of getting a black pair:
12/30 * 11/29 = 132/870
Following the same process for the other colors gets you:
P(matching white) = 8/30 * 7/29 = 56/870
P(matching blue) = 6/30 * 5/29 = 30/870
P(matching red) = 4/30 * 3/29 = 12/870
Adding the probabilities gets a total of 230/870 = 23/87 or about 26.4%.
In this case you need to remember that when there are two individual events that must BOTH occur, you multiply their probabilities together. When there are multiple desired outcomes that are EACH satisfactory you add their probabilities.