1. Essential Ideas for The Nature of Probability

2. An event is an outcome from an experiment
An event is an outcome from an experiment. An experiment is any process that can be repeated in which the results are uncertain.
The probability of an event is a measure of the likelihood of its occurrence. The long-term proportion with which a certain outcome is observed is the probability of that outcome.
A probability model lists the different outcomes from an experiment and their corresponding probabilities
To construct probability models, we must know the sample space of the experiment. The sample space S lists all the possible outcomes of an experiment.
For example, the sample space resulting from the experiment of rolling a die is:
S = {1, 2, 3, 4, 5, 6}

3. Properties of Probabilities
For a sample space S:
The probability of each event in S occurring is between 0 and 1.
The sum of the probabilities of all events in S occurring is equal to 1.
If an event is impossible, the probability of the event is 0.
If an event is a certainty, the probability of the event is 1.

4. There are three methods for determining the probability of an event:
1. The classical (theoretical) method
The empirical (experimental) method
The subjective method

5. Classical Method
The classical method of computing probabilities requires equally likely outcomes, meaning that each event has the same probability of occurring. The corresponding theoretical probabilities are obtained by logical reasoning according to stated definitions.

6. Computing Probability Using the Classical Method
If an experiment has n equally likely simple events and if the number of ways that an event E can occur is s, then the probability of E, denoted P(E), is:
Where s = number of ways that E can occur and
n = Total number of possible outcomes

7. Examples of theoretical probabilities
Suppose that a single card is drawn from an ordinary deck of 52 cards.
Then,
P(ace) = 4/52 = 1/13 ≈ 0.08
P(heart) = 13/52 = 0.25
P(face card) = 12/52 = 3/13 ≈ 0.23

8. Suppose you are beginning a game of Monopoly
Suppose you are beginning a game of Monopoly. What is the probability that you land on a railroad on your first roll of the dice? The sample space for rolling a pair of dice is given below.

9. Since you roll a pair of dice in the game of Monopoly, you must determine how many spaces there are to the first railroad. (It is impossible to get to the other three railroads on the first roll, because they are more than 12 spaces away.)
You must move five spaces on your first roll of the dice to land on the first railroad.
Looking back at the sample space, there are four ways to obtain a combined roll of “5”, so the probability of landing on a railroad on your first roll of the dice is:
P(five) = 4/36 = 1/9 = 0.11

10. Empirical Method
Empirical probabilities are obtained from experimental data.
As an experiment is repeated more and more times, the empirical probability will tend to come closer and closer to the theoretical probability of that event.

11. Computing Probability Using the Empirical Method
The probability of an event E is approximately the number of times E is observed divided by the number of repetitions of the experiment.
For example, suppose that in a certain study, 46 out of 155 people showed a certain kind of behavior.
P(behavior shown) = 46/155 ≈0.30
This is an empirical probability because it was arrived at through experimentation.

12. Law of Large Numbers
The theoretical probability of rolling a die and obtaining a “3” is 1/6 because there are six possible outcomes, each with an equal chance of winning. If we roll a die 100 times and obtain a “3” on twenty rolls, the empirical ( or experimental) probability is 20/100 = 1/5. As we continue to roll the die, the empirical probability will tend to come closer and closer to the theoretical probability of 1/6. This is an example of the Law of Large Numbers.

13. Subjective Method
Subjective probabilities are probabilities based upon an educated guess.
For example, there is a 30% chance of rain tomorrow.

14. Probabilities of Unions and Intersections
Let E and F be two events.
P(E U F) is the probability that either E or F occurs
P(E ∩ F) is the probability that both E and F occur.

15. Suppose that a single card is drawn from an ordinary deck of card.
What is the probability that it is a two or a king?
P(two U king) = (4 + 4)/52 = 8/52 = 2/13 ≈ 0.15
What is the probability that it is a two or a heart?
P(two U heart) = (4 + 12)/52 = 16/52 = 4/13 ≈ 0.31
What is the probability that it is a two and a heart?
P(two ∩ heart) = 1/52 ≈ 0.02
What is the probability that it is a two and a king?
P(two ∩ king) = 0/52 = 0

16. Independent Events
Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F.
Two events E and F are dependent if the occurrence of event E in a probability experiment does affects the probability of event F.

17. Independent Events and Intersection
If events E and F are independent events, then we can find the probability of an intersection using the multiplication property of a probability:
P(E and F) = P(E ∩ F) = P(E)·P(F)

18. Example: Suppose a coin is tossed and a die is simultaneously rolled.
What is the probability of tossing a tail and rolling a “4”?
P(tail ∩ 4) = (1/2)·(1/6) = 1/12

19. Independent Events and Union
If events E and F are independent events, then we can find the probability of a union using the addition property of probability:
P(E or F) = P(E U F) = P(E) + P(F) – P(E ∩ F)
= P(E) + P(F) – P(E)·P(F)

20. Example: Suppose a coin is tossed and a die is simultaneously rolled.
What is the probability of tossing a tail or rolling a “4”?
P(tail U 4) = P(tail) + P(4) – P(tail ∩ 4)
= 1/2 + 1/6 – (1/2)(1/6) = 7/12 ≈ .58

21. Mutually Exclusive Events
If events E and F have no simple events in common or cannot occur simultaneously, they are said to be mutually exclusive or disjoint. The addition rule for mutually exclusive events simplifies to:
P(E U F) = P(E) + P(F)
because P(E ∩ F) = 0 if the events are mutually exclusive.
Note: Mutually exclusive events and independent events are not synonymous!

22. Complementary Probabilities
Let
s = number of ways an event can occur (successes)
f = number of ways the event cannot occur (failures)
n = total number of possible outcomes
so that (s + f) = n
Then, the probability that event E occurs is P(E) = s/n
The probability that event E does not occur is = f/n
so that P(E) + = 1
Two important properties of complements are:

23. What is the probability of obtaining at least one head in three flips of a coin?
We could answer this with a tree diagram, but it is easier to use the complement along with the multiplication rule.
The complement of this problem is to determine the probability of obtaining no heads in three flips of a coin. This is the same thing as determining the probability of obtaining all tails in three flips of a coin. We can use the multiplication property of a probability to answer this question:
P(tail ∩ tail ∩ tail) = (1/2)·(1/2)·(1/2) = 1/8
So, the probability of obtaining at least one head is:
1 – 1/8 = 7/8

24. Conditional Probabilities
A conditional probability is a probability of an event given that another event has occurred.
We denote this by:
Read this as: “probability of E given F”
A conditional probability will alter the sample space of E.

25. A single card is drawn from a standard deck of cards
A single card is drawn from a standard deck of cards. Find the following conditional probabilities.
The probability of drawing a face card given that the card drawn is a jack is 1 (since all jacks are face cards).
The probability of drawing a jack given that the card drawn is a face card is 4/12 = 1/3 because there are 12 face cards in a standard deck and 4 of them are jacks.

26. Drawing with Replacement
Drawing with replacement means choosing the first item, noting the result, and then replacing the item back into the sample space before selecting the second item. With replacement, the events are independent.
Find the probability of drawing a spade on the first draw and a heart on the second draw with replacement.

27. Drawing without Replacement
Drawing without replacement means choosing the first item, noting the result, and then selecting a second item without replacing the first item.
Find the probability of drawing a spade on the first draw and a heart on the second draw without replacement.

28. Expected Value / Expectation
A game is said to be fair if the expected value is 0. If the expected value is positive, then the game is in your favor. If the expected value is negative, then the game is not in your favor. The expected value (or your expectation) of winning is computed using the following formulas:

29. Expected Value Formulas
If there is no cost to play:
Expectation = (Amt. to win) x (Prob. of winning)
If there is an “up front” cost for playing the game, then:
Expectation = (Amt. to win) x (Prob. of win) – Cost of playing
If there is a “leave your money on the table” cost, then:
Expectation = (Amt. to win) x (Prob. of win) – (Cost to play)(Prob. Of losing)

30. Suppose you draw a card from a deck of cards and are paid $10 if it is an ace. Find the following expected values.
If there is no cost:
Expectation = $10 x (4/52) ≈ $0.77
If the same game costs you $1 “up front” to play:
Expectation = $10 x (4/52) – $1 ≈ –$0.23
If the same game costs you $1 but you leave your money “on the table” (if you win you get your $1 back).
Expectation = $10 x (4/52) – $1(48/52) ≈ –$0.15

31. Mathematical Expectation
If an event E has several possible outcomes with probabilities , and if for each of these outcomes the amount that can be won is
, then the mathematical expectation (or expected value) of E is:
Expectation

32. Example
Suppose you play a game in which you have a chance to win either $1, $5, $10, $20, or $100 depending on which bill you draw out of the box. There are ten $1 bills, four $5 bills, three $10 bills, two $20 bills, and one $100 bill. You pay $20 up front to play the game. What is the expected value? Should you play this game?
Expected value = $1(10/20) + $5(4/20) + $10(3/20)
+ $20(2/20) + $100(1/20) - $20
= -$10.00
You should NOT play this game!

33. Odds Let s = number of ways an event can occur
f = number of ways an event cannot occur
n = total number of possibilities
Then, the odds in favor of an event E = s/f
(the ratio of the number of ways the event can occur to the number of ways it cannot occur)
And the odds against an event E = f/s
(the ratio of the number of ways the event cannot occur to the number of ways it can occur)

34. Example
If a jar has 2 quarters, 200 dimes, and 800 pennies, and a coin is chosen at random, what are the odds against picking a quarter?
Since there are 1000 ways of not picking a quarter and there are only 2 ways of picking a quarter, the odds against picking a quarter (meaning the odds that you pick something besides a quarter) = 1000/2 = 500/1
We say the odds against picking a quarter at random are 500 to 1.

35. Finding the Odds Given the Probability
Let P(E) denote the probability of an event occurring. Then the
odds in favor of E =
odds against E =

36. Finding the Probability Given the Odds in Favor or the Odds Against an Event
If you know the odds in favor of an event E (the ratio of s to f ), or if you know the odds against an event E (the ratio of f to s), then
and

37. If the probability of an event is 0
If the probability of an event is 0.45, what are the odds in favor of the event?
Since P(E) = 0.45 = 45/100 it follows that the number of ways the event can occur is 45. This implies that the number of ways the event cannot occur is 55 (because = 100 which is the total number of possibilities indicated by the value in the denominator).
Thus, the odds in favor of E are 45 to 55 which reduces to 9 to 11.

38. If the odds against you are 20 to 1, what is the probability of the event?
20 represents the number of ways the event cannot occur.
1 represents the number of ways the event can occur.
This implies that there are 21 total possibilities, so the probability that the event occurs = 1/21.