Probability Models
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)
How do you convert a value to a z-score?
z = (x - mu)/sigma
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
What must be true for an event to create a binomial probability distribution?
DOUBLE JEOPARDY: What is the formula for P(X = k) for k successes in n trials?
1. Each trial has two outcomes (success and failure)
2. The probability of success on each trial is the same
3. Trials are independent.
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
What is expected value and how do you find it?
Expected value is the mean of a probability function. Add up the products of X and P(X).
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
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?
10