Probability Key Questions
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.
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
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.
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.
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.
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.
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
What is a continuous random
variable? Give an example.
• A continuous random variable is a random
variable that maintains its ‘randomability’
throughout
What is conditional probability?
• Conditional probability is the probability
that an event will occur given that another
event has happened
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.
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)
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
What is the mean of a geometric
distribution?
• The mean of the geometric distribution is
equal to 1/p
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
What is the standard deviation of a
geometric distribution?
• There is no standard deviation for
geometric distributions.
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.
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.
For a proportion problem, when is
the standard deviation at its
largest?
• The standard deviation is at its largest
when the probability is .5
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.
What is replacement and nonreplacement?
• 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
Nonreplacement
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).
Complement. What is it?
• If the probability of an event happening is
P, then the complement of P is [1-P]
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
Pay 0
out
1
2
3
P(X)*payout+ P(X)*payout+ P(X)*payout…
.25(0)+.5(1)+.20(2)+.05(3)=
.5+.4+.15=1.05
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
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).