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Introduction

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.
If you don't know the answers to the questions in the first two examples, the answers are given at the end of this section. But the answers for examples (3),(4) and (6) are not that easy to get.

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

Examples:
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 }.
But in terms of probability it is not the sample space that is of basic interest, but subsets of it.

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 Ac .


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.


Exercise 3.1.1  
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.


next up previous contents index
Next: Probabilities Up: Probability Previous: Probability

2001-01-01