9/30/10 Notes

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LSP 121 -- 9/30/10

Probability

The probability that an event occurs is a value between 0 and 1 that indicates how likely that event is to occur. (0 means that the event is impossible, 1 means that the event is certain.)
Examples of events:
Three common ways of determining probabilities:
1. Theoretical or A Priori Probability
  Assume that all outcomes are equally likely. Suppose you are interested in the probability that the event A occurs.
  Example 4: What is the probability of getting one head out of two coin flips?
  Ans: The set of all outcomes = {HH, HT, TH, TT}
  A = Event of getting exactly one head = {HT, TH}
  P(A) = # of events in A / # events in S = 2 / 4 = 0.5
2. Empirical Probability
  Perform a repeated experiment. Count the number of times s the event A occurs and the total number of times n the experiment is repeated.
  Example 4: Flip a coin 1 million times. 499,523 heads were obtained. The probability of obtaining a head is 499,523 / 1,000,000 = 0.499523.
3. Subjective Probability
  Each person uses personal intuition or judgement.
  What is the probability that I will need an umbrella today?
  Las Vegas oddsmakers combine subjective probabilities for many sources when determining betting odds.
  Example 5: What is the probability that the Bears will win the Super Bowl in 2011?
4. Fair Bets
  The concept of a fair bet is sometimes used to determine subjective probabilities. A fair bet is a bet for which you would be willing to take either side.
Rules for Probabilities:
1. Every event has a probability is between 0 and 1, inclusive, of occuring.
2. An event with probability 1 is certain to occur.
3. An event with probability 0 is impossible.
4. If p is the probability that the event A occurs, then 1 - p is the probability that A does not occur.

Random Variables

A random variable is the process of choosing a random number.
Example 6: Pick a person at random. That person's height is a random variable.
Example 7: The random variable x is the number of heads obtained out of two coin flips.

x Probability

0 0.25

1 0.50

2 0.25
Example 8: The random variable x is the number of heads obtained out of three coin flips.

x Probability

0 0.125

1 0.375

2 0.375

3 0.125

x	Probability
0	0.25
1	0.50
2	0.25

x	Probability
0	0.125
1	0.375
2	0.375
3	0.125

The Risk of a Random Variable

The risk of a random variable is the amount that you expect to win or lose for each outcome of the random variable.
Another term for the risk is the expected value.
The risk of a random variable is its theoretical average, based on the probabilities of its outcomes.

The risk value of a random variable x can be calculated using the short description table.

x Probability

x₁ p₁

x₂ p₂

... ...

x_n p_n

Expected value = E(x) = x₁ p₁ + x₂ p₂ + ... + x_n p_n

The expected value is the weighted average of the outcomes of the random variable. You don't need to divide by the sum of the weights because the sum of the probabilities is 1.
Example 9: Suppose the probability that the Bears will the Super Bowl next year is 1 / 5 = 0.2. You make a bet with your friend. If the Bears win you get $100. If the bears do not win, you pay him $30. What is the expected value of this bet?

Payoff Probability

-30 0.8

100 0.2

Note: the payoff of 30 in the table is negative because you are losing $30.
Expected value = (-30) × 0.8 + 100 × 0.2 = -0.4
The conclusion is that you will lose 40 cents every time you make this bet.
Example 10: A tropical island has the same probability distribution for rain every day:

Rainfall Probability

0 0.3

1 0.4

2 0.2

3 0.1

What is the risk (expected value of rain today)?
Ans: E(x) = x₁ p₁ + x₁ p₁ + x₁ p₁ + x₁ p₁
= 0 × 0.3 + 1 × 0.4 + 2 × 0.2 + 3 × 0.1 = 1.1
Example 11: Your house costs $400,000, on which you pay $130 per year in insurance. If the probability that your house is destroyed in a given year is 0.025%,
1. what is the risk of the insurance company for your house in a given year.
  Ans: The insurance company gains $130 if your house is not destroyed, but loses $400,000 if it is:
  Risk = 130 * 0.99975 * 130 + (-400,000 + 130) * 0.00025 = -99.87.
2. what is your risk in a given year?
  Ans: You lose 130 per year regardless of what happens to your house.
3. what is your risk in a given year if you don't have insurance.
  Ans: Risk = 0 * 0.99975 + (-400,000) * 0.00025 = -100
4. what is the "pure premium" for the insurance company, that is what would your cost of insurance be if the risk to the insurance company for your house were 0?
  Ans: If the risk is 0 and the premium is p,
  Risk = p * 0.99975 + (-400,000) * 0.0025 = 0. Solve for p: p = 100.025.

Payoff	Probability
-30	0.8
100	0.2

Rainfall	Probability
0	0.3
1	0.4
2	0.2
3	0.1

The Multiplication Rule for Independent Events

Two events A and B are independent if the occurence of A does not affect the probability that B occurs.
Example 12: Flip a coin twice. Define the event A to be getting a head on the first toss. Define the event B to be getting a head on the second toss. The events A and B are independent.
The Multiplication Rule: If A and B are independent, then P(A and B) = P(A) × P(B).
Example 13: Flip a coin 10 times. What is the probability of getting all heads?
Example 14: The probability that I will die in a single plane ride is 0.0001. What is my probability of dying in 500 independent plane crashes?
We can also use this formula to compute the probability of at least one success in n independent tries:
Assume that the probability of success in one try is p.
Then the probability of dying in 500 independent plane crashes is
Example 15: Roll a pair of dice 24 times. What is the probability of obtaining snakeeyes (pair of ones) at least once?
Ans: 1 - (1 - p)ⁿ = 1 - (1 - 1/36)²⁴ = 0.491414

Activity 7

Work on Activity 7.