What does “not” mean
subtract from 1. i.e.: if p(blue eyes) = 70%, then P(not blue eyes) = 1 - .7 = 30%
what is the law of large numbers?
Theorem that tells us that as the number of independent trials increase, that the results will become closer and closer to the theoretical probability
law of averages
misunderstanding of the law of large numbers: thinking that if a good hitter has struck out 3 times that he is “due” for a hit. The law of large numbers works only in the long run, not in the short term.
P(A) = (# times A occurs)/(total # trials) in the long run
first three rules for working with probability
1, make a picture
- make a picture
- make a picture
If S = your sample space, then
events that have no outcomes in common. for example, choosing a freshman, and choosing a sophomre from a mixed group of students would be disjoint events.
Addition Rule (disjoint events)
for two disjoint events, A and B, the probability that one OR the other occurs is the sum of the probabilities of the two events.
P(A U B) = P(A) + P(B), A and B disjoiont
Multiplication Rule (independent events)
If two events A and B are independent, that the probability of both A AND B occurring is the product of the two probabilities
P(A and B) = P(A) times P(B), A and B independent
General Addition Rule:
P(A or B), events not disjoint
P(A) + P(B) - P(A and B)
Are the events “heart” and “black jack” disjoint? independent?
“heart” and “black jack” have no common outcomes (there are no black jack of hearts!) so they are disjoint. However, they are not independent. P(black jack) = 2/52, but the P(black jack/lheart) = 0!
What is expected value?
The mean. What you would average if you played the game A LOT!
What is the mean of a random variable?
The expected value. The sum of the probabilities times the values.
Can disjoint events be independent?
NO! IF the are disjoint, then knowing one tells you that the other could not happen. Therefore they are NOT INDEPENDENT
You own a tire shop, and order tires from two companies, A and B. 80% from A and 20% from B. 1% of tires from A are defective, and 4% from B are defective. What strategy would you use to determine the probability that a defective tire comes from A?
Make a tree diagram!
What is the probability that your first success occurs on the 7th try?
What is a probability model?
All possible values of the random variables with their respective probabilities.
When combining probability models, do you add or subtract standard deviation?
Neither! Square the standard deviations, add them, then take the square root.
What is the probability of at least one?
1 - (probability of none)
What does the geometric model tell us?
About the first success. Example: what is the likelihood the first success is on the fifth trial?