We need some, not much, **probability** for this class. It will help with assessing noise in samples. But, also, we can solve some very interesting problems in a simple fashion. In order to look at such problems several years ago, we would have had to stop and develop some mathematics. With **resampling** we no longer have to do this.

Consider first some simple examples:

- 1.
- Flip a
*fair*coin. What's the probability of a head? - 2.
- Roll a
*fair*6-sided die. You win the game if a 1 or 2 is the upface. What's the probability that you win? - 3.
- Roll a pair of
*fair*6-sided dice. You win the game (on the first roll) if the sum of the upfaces is 7 or 11. What's the probability that you win? We will refer to this as the game craps in subsequent text. - 4.
- Five cards are dealt from a standard
*well shuffled*deck of 52 cards. What's the probability that the hand contains a pair. That is, what's the probability that in five card poker you open with a pair? - 5.
- In a simple lotto you pick a number from 1 to 50. Later, to determine the winner, one number is selected at random. Find the probability that you win.
- 6.
- In Lotto 2, you select 4 numbers from the numbers 1 through 50. Find the probability that you win.

We need a little nomenclature here which easily leads to the solution for (3) and will help contemplate the solution for (4) and (6).

- An
**experiment**results in an outcome. - The collection of all outcomes is the
**sample space**. We shall denote sample spaces with the letter*S*.

- 1.
- Flip a coin:
*S = {H,T}.* - 2.
- Roll a six sided die:
*S={1,2,3,4,5,6}.* - 3.
- Roll a pair of 6-sided dice:
*S = {(1,1),(1,2), (1,3), ..., (6,6)}*. That is,*S*consists of 36 pairs of integers. Here's a picture of*S*: (Read the points as (Die 1, Die 2).)6 - * * * * * * Die 2 - - 5 - * * * * * * + - - 4 - * * * * * * - + 3 - * * * * * * - - 2 - * * * * * * + - 1 - * * * * * * ----+---------+---------+---------+---------+---------+--Die 1 1 2 3 4 5 6

- 4.
- Five cards are dealt from a standard deck of 52 cards. I don't think I'll list
*S*in this case since it contains over 2 and half million elements. - 5.
- Play simple lotto :
*S = {1,2, ..., 50}*. - 6.
- Play Lotto 2 :
*S = { all subsets of 4 numbers drawn from 1 through 50 }*.

An **event** is a subset of S. Denote events by *A*, *B*, *C*, etc. We say the event *A* **occurs** if the experiment results in an outcome in *A*; i.e., *A* comes up. The **complement** of the event *A* occurs if *A* does not occur. We will sometimes write the complement of *A* by *A*^{c} .

**Examples**

- 1.
- Flip a coin:
*A={H}*. - 2.
- Roll a six sided die:
*B={1,2}.* - 3.
- Roll a pair of 6-sided dice:
*A= sum of upfaces 7 or 11*. Find*A*on the picture:6 - * * * * * * Die 2 - - 5 - * * * * * * + - - 4 - * * * * * * - + 3 - * * * * * * - - 2 - * * * * * * + - 1 - * * * * * * ----+---------+---------+---------+---------+---------+--Die 1 1 2 3 4 5 6

Note that the 7's (sum of upfaces 7) fall along the main diagonal starting with the point (1,6) and ending with the point (6,1). - 4.
- Five cards are dealt from a standard deck of 52 cards :
*A= just a pair*. Alas,*A*is too big to list, also, because it has over a million elements. But here is one element:*{Jack of hearts, jack of clubs, 7 of diamonds, 9 of spades, 2 of clubs}*. What's another such hand? - 5.
*A*is the event that you picked the winning number in the simple lotto.- 6.
- (a)
*A*is the event that you picked the 4 winning numbers in Lotto 2.- (b)
- You buy 100 Lotto 2 tickets.
*B*is the event that one of your tickets is the winner.

**Answers :**

- 1.
- The probability of a head on the flip of a
*fair*coin is 1/2. - 2.
- The probability of getting a 1 or 2 on a roll of a fair 6-sided die is 2/6 = 1/3.

- 1.
- List the sample space, list the event of interest, and its complement for the experiment: Spin a spinner with the numbers 1 through 10 on it. Suppose we are interested in the event an odd number spun.
- 2.
- List the sample space, list the event of interest, and its complement for the experiment: Roll a pair of 6-sided dice. We are interested in the event that both dice are the same.
- 3.
- List the sample space, list the event of interest, and its complement for the experiment: A pizza can have none, one, two or three of the toppings onions, extra cheese, or peppers. We are interested in a pizza with only two toppings.
- 4.
- List the sample space, list the event of interest, and its complement for the experiment: From a standard deck of 52 cards, three cards are dealt (without replacement) and their color is observed. We are interested in getting 3 red cards.