1. Probability Key Questions

2. What does independent mean?
Independent events are events that have no effect on one another regardless of the outcome
Ex. Flipping 2 coins. The probability of Coin 2 yielding heads is unaffected by whether or not Coin 1 yields a heads or not.

3. What does mutually exclusive mean?
Mutually exclusive events cannot happen at the same time.
Ex. You cannot get both a heads and a tails on a single coin at the same time, thus the events are mutually exclusive

4. Which of the previous has to do with and (multiply) problems?
Independence has to do with “and” problems.
If two events are independent, the chances of both of them occurring is the probability of both events multiplied together.

5. Which of the previous has to do with or (addition) problems?
Mutually exclusive problems have to do with “or” problems.
If two events are mutually exclusive, you can add their probabilities to find the probability of one or the other occurring.

6. How do you find the mean of a discrete random variable?
The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take.

7. How do you find the standard deviation of a discrete random variable?
The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.

8. What is a discrete random variable?
A discrete random variable is a variable that can be “counted” such as the number that appears on a dice after it is rolled.
An example of a variable that is not discrete is height

9. What is a continuous random variable? Give an example.
A continuous random variable is a random variable that maintains its ‘randomability’ throughout

10. What is conditional probability?
Conditional probability is the probability that an event will occur given that another event has happened

11. What is the mean of a binomial distribution?
The mean and variance of the binomial distribution are equal to the sum of the means and variances of the n independent Z variables.

12. What is the standard deviation of a binomial distribution?
. The standard deviation is the square root of the variance.
------
To find variance of a binomial distribution,
np (1-p)

13. What are the conditions for a binomial?P
The probability of the event remains the same for each trial O
There are two possible outcomes: it either happens or it doesn’t T
The number of trials I
Each trial must be independent

14. What is the mean of a geometric distribution?
The mean of the geometric distribution is equal to 1/p

15. What are the conditions for a geometric distribution?P
The probability of the event remains the same for each trial O
There are two possible outcomes: it either happens or it doesn’t I
Each trial must be independent

16. What is the standard deviation of a geometric distribution?
There is no standard deviation for geometric distributions.

17. What is the formula for combining standard deviations?
The formula for combining standard deviations is squaring each standard deviation and adding them together. Then, take the square root of that sum.

18. What is a standard score?
A standard score indicates how many standard deviations an observation or datum is above or below the mean. May also be referred to as z-score.

19. For a proportion problem, when is the standard deviation at its largest?
The standard deviation is at its largest when the probability is .5

20. How do you find the median of a discrete random variable?
The median of a discrete random variable is the "middle" value. It is the value of X for which P(X < x) is greater than or equal to 0.5 and P(X > x) is greater than or equal to 0.5.

21. What is replacement and non-replacement?
Replacement indicates that the probability of an event occurring remains the same no matter how many trials occur
Non-replacement indicates that with each trial the probability of an event changes.
Ex: Drawing a face card
Replacement
Non-replacement
16/52 chance that a face card is drawn throughout trial
16/52 for first trial
For each trial the probability is updated. So, given the first card is a face card, the second trial would have a 15/51 chance of drawing a face card (or 16/51 if not).

22. Complement. What is it?
If the probability of an event happening is P, then the complement of P is [1-P]

23. How do you calculate payout?
Multiply the probability of an event occurring by the amount it pays out and add results.
P(X)
.25
.50
.20
.05
Payout 1 2 3
P(X)*payout+ P(X)*payout+ P(X)*payout…
.25(0)+.5(1)+.20(2)+.05(3)=
=1.05

24. What is the law of large numbers?
The Law of Large Numbers states that as n goes up, the frequency of events will more closely resemble their actual probability.
Ex. The more you flip a coin, the closer to average it would be; it’d get closer and closer to 50% heads and 50% tails

25. What are the degrees of freedom for each test we run?
For tests for means degrees of freedom is n-1.
For linear regression tests, the degrees of freedom is n-2.
For chi-square tests, the degrees of freedom is (rows-1) x (columns-1).