# Probability and Statistics

### Topics

## Introduction to :

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 two 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 sucking on 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 are 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

.

#### Example 1

A drawer contains 10 red socks and 8 green socks. If two socks are randomly drawn from the drawer, what is the probability of getting a pair of red socks? |

#### Exercise 1

A jar contains 3 red, 2 blue, and 8 green marbles.

- If one marble is drawn at random from the jar, what is the probability that marble is green?
- If two marbles are drawn at random from the jar, with the first marble being replaced before the second is drawn, what is the probability that both marbles are green?
- If two marbles are drawn at random from the jar and the first marble is not replaced before the second is drawn, what is the probability that both marbles are green?
- What is the probability of you losing all your marbles?

#### Exercise 2

Mr. and Mrs. Kappler bought ten books and plan to give one at random to each of their five kids. No one ever said they were attentive or considerate parents. Certainly not their children. If there is only one such way to distribute the books that will make every kid happy with their book, what is the probability that every kid will be happy with his or her book?

#### Exercise 3

An ice cream parlor has 30 flavors but Su likes only half of them. It isn't because she's picky. This parlor features the flavors Charred Chocolate Chip, Swine Swirl and Deuce de Leche. It doesn't actually do a lot of business.

If the scooper makes a three-scoop ice cream cone at random (using three different flavors),

(a) What is the probability Su will like all three flavors?

(b) What is the probability Su will like at least one flavor?

#### Exercise 4

A jar has 10 vanilla candies, 4 chocolate candies, and 1 strawberry candy. Seems that shrimp candy company went under. Shocker. If two candies are drawn at random from the jar without replacement, what is the probability that both are strawberry?