Math 37 - Lecture 12

Probability cont.

If the probability of someone carrying a bomb on a plane is .001, then the chance of two people carrying a bomb is .000001. Therefore, I should start carrying a bomb on every flight?

Example Common sources of caffeine are coffee, tea, and cola drinks. Suppose 55% of adults drink coffee, 25% of adults drink tea, and 65% of adults drink cola.

- The probability a randomly selected adult drinks coffee or cola?

- What is the probability two randomly selected adult drink cola?

Def:Random variable = variable whose random value is numerical.

Examples:

X=number of heads in 4 coin tosses

Y=height of a person randomly selected from population

Z=number of boys in 3 children

Random variables also have probability distributions.

List all possible outcomes of the random variable and their probability of occurring.

Types of Random Variables

1. Continuous = variable can take any value in an interval

2. Discrete = variable has a finite number of possible values

Example The probability a couple has 2 boys out of 3 children?

Random variable? Discrete or Continuous? Prob Distribution?

Example Roulette: Your friend bets one dollar each time he plays. There are 38 numbers:18 red, 18 black, and 2 green (00). If he wins, he gains a dollar. If he loses, the dollar is gone. Let X=amount of money he wins on each play.

Probability Distribution:

How do you decide if this is a good bet? What pieces of information are important?

First ten games: win, win, lose, win, win, win, lose, win, lose, lose

Should he keep playing?

Expected Value of a discrete random variable

E(X)= SxiP(X=xi)

What is your friend's expected winnings?

 

Example Let X=number of heads in 4 coin tosses. What is the expected number of heads?

 

How do Casinos make money? Insurance Companies?

Law of Large Numbers

If we observe the outcomes of a random variable over and over again, and calculate the average of the observed values, this average gets closer and closer to the true mean of the distribution.

Can also define the variance of a discrete random variable:

Var(X) = Si(xi - EX)2pi

Why do we need to know about random variables?

Let = proportion of heads in 20 tosses of a fair coin.

Is the outcome of known?

Does the outcome of follow a pattern?

is a random variable!

Def: Parameter = number that describes the population.

Statistic = number that is computed from data in a sample

Example: A poll at UOP asks which presidential candidate a student favors. If we could ask everyone, we'd find 57% favor Candidate A. In a sample of 50 student, 49% favor Candidate A.

Parameter =

Statistic =

Typically, the value of the parameter is unknown. We take one sample and calculate a statistic and use that to estimate the parameter.

Next Time: Section 3.4 - very important

Possible Extensions Topic: I have a paper discussing expected winnings for several casino games.