Probability Models
What are the two characteristics of a valid probability model?
1. All the probabilities need to be from 0-1 inclusive.
2. The sum of the probabilities needs to be 1.0
What must be true to create a binomial probability model?
It must be a Bernoulli Trial
When is it appropriate to use a normal model to approximate a binomial distribution?
If we expect at least 10 successes and 10 failures (or np >= 10 and nq >= 10)
The notation for the Expected Value, how you would represent it.
What is E(X), 'E of X'.
Is the following probability model valid? Why?
P(X=0):0.72
P(X=1):0.21
P(X=2):0.02
P(X=3):0.03
No, the probabilities do not add to 1.0
What 4 things must be true for a situation to be a Bernoulli Trial?
Set probability for success.
Events must be independent.
A set number of trials are going to be performed.
How do you convert a value to a z-score?
z = (x - mu)/sigma
The expected value means...
The average value for the probability model.
What must be true for a situation to be a probability model?
1. Random variable (outcomes are assigned a number)
2. Each outcome has a probability
3. The probabilities add to 1
A six sided die is rolled three times. What is the probability that you roll a three on 2 of the rolls.
Is this a Binomial Probability?
DOUBLE JEOPARDY: What is the formula for P(X = k) for k successes in n trials?
Yes
Formula: (nCk)*p^k*q^(n-k)
Which calculator function do you use to find the area underneath the normal curve? Which calculator function do you use to find the z-score for a certain area?
Normalcdf
invNorm
How do you find it?
Add up the products of X and P(X) or multiply each value of x by it's probability and add them together.
We will roll a single 6-sided die. Let random variable X = the number that comes up. Explain why the probability model for X is NOT binomial.
DOUBLE JEOPARDY: In the context of rolling the die, define a random variable that would have a binomial probability model.
There are 6 events possible for the random variable and would need to define only two so that there is success or failure.
Ex: Rolling an even is success and rolling odd is failure.
What are the short cuts (formulas) to finding the expected value and standard deviation for a binomial distribution?
E(X) = np
sigma = sqrt(npq)
Find the Normal probability P(z < -0.6).
0.274
Suppose that 23% of all college students are married. Out of a random sample of eight college students how many would you expect to be married?
about 2 or 1.84
In basketball, a play shooting "one-on-one" takes one foul shot, and if they make the first one, they can shoot a second shot. Suppose that, overall, this player makes 80% of his foul shots.
Create a probability model for the number of points the player may score on a one-on-one opportunity.
N 0 0.2
YN 1 0.16
YY 2 0.64
You roll four dice. Find the probability of at least two 6's.
0.132
A certain type of tire will last for an average of 37,200 mi, with a standard deviation of 2650 mi. Suppose that the tread life can be described by a Normal model. For how many miles can the company guarantee these tires, if the company wants at least 90% of them to last that long?
33,800 mi
A game involves earning points by rolling a special 6-sided die that says 5 on three faces, 10 on two, and 25 on the sixth face. What's the expected number of points a player will earn by rolling this die? What is the standard deviation?
10 and 7.071