More Probability at a Glance

Now that we can find numbers of permutations and combinations, we can find more complicated probabilities.

Sample Problem

A jar contains 5 vanilla candies, 9 spice candies, and 3 shrimp candies. That's what you get when you go junk food shopping in eastern Maine. What is the probability of drawing 2 vanilla candies at random from the jar, without replacement?

There are 17 total candies. There are  possible ways to choose 2 candies from 17 (order isn't important; we don't care which vanilla candy we draw first, as long as we aren't grabbing anything seafood-flavored). The number of possible outcomes is .

How about the number of favorable outcomes? In this case, an outcome is favorable if we pick 2 vanilla candies. It doesn't matter which two vanilla candies we pick, and there are 5 to choose from, so the number of favorable outcomes is .

Simplifying, we see that

and

The final probability is

.

The probability that we'll need to dip our candies in cocktail sauce? Let's hope close to zero.

Sample Problem

A drawer contains 4 blue socks, 3 red socks, and 5 green socks. The dryer captured a couple victims during its last cycle, hence the odd numbers. Two socks are drawn at random from the drawer (without replacement). Find the probability of getting

(a) a pair of red socks.

(b) a pair of blue socks.

(c) a pair of socks that are the same color.

Answers:

(a) 

(b) 

(c) A pair of socks that are the same color can be blue, red, or green. These are mutually exclusive (a pair of socks can only be one color at once, even if they're mood socks that change color to show the type of day your feet are having) so the probability of getting a pair of socks of the same color is

(probability of a red pair) + (probability of a blue pair) + (probability of a green pair).

The only one of those probabilities we haven't already found is the probability of a green pair, but finding that is exactly like parts (a) and (b) of this problem:

To add up all the probabilities, let's get our kicks using the fractions with denominator 66. The probability of getting a pair of socks of the same color is

(probability of a red pair) + (probability of a blue pair) + (probability of a green pair)

or

.