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8.1
Binomial Distribution and Probability
The Binomial Setting
1. Each observation falls into one of just two categories, which for convenience we
“success” and “failure.”
2. The probability of a success, call it p, is the same for each observation.
3. The observations are all independent.
4. There are a finite number of observations.
called
Mean and Standard Deviation of a Binomial Random Variable
-If X is a binomial random variable with probability of success p on each trial, then the mean, or
expected value, of the random variable is x = np.
-The standard deviation of X is √np(1-p)
Binomial Equation (*Important to show on CollegeBoard Free-Response)

P(X = k) = ( )pk(1-p)n-k
* “At least” indicates the need for 1 - ( )pk(1-p)n-k or 1- binomcdf on the calculator
8.2
Geometric Distribution and Probability
Geometric Variable: a random variable X that is defined as the number of trials needed to
obtain the first “success.”
Geometric Distribution: distribution produced by a random geometric variable.
 Flip a coin until you get a head.
 Roll a die until you get a 3.
 In basketball, attempt a three-point shot until you make a basket.
The Geometric Setting
1. Each observation falls into one of just two categories, which for convenience we called
“success” and “failure.”
2. The probability of a success, call it p, is the same for each observation.
3. The observations are all independent.
4. The variable of interest is the number of trials required to obtain the first success.
Rule for Calculating Geometric Probabilities
If X has a geometric distribution with probability p of success and (1-p) of failure on each
observation, the possible values of X are 1, 2, 3, 000 If n is any one of these values, the probability that
the first success occurs on the nth trial is
P(X = n) = (1-p)n-1p
Geometric Probability Distribution
Value of X
1
2
Probability
p
(1-p)p
3
(1-p)2p
…
…
Mean and Standard Deviation of a Geometric Random Variable
-If X is a geometric random variable with probability of success p on each trial, then the mean,
or expected value, of the random variable, that is, the expected number of trials required to get the
first success, is μ = 1 / p.
-The variance of X is (1 – p) / p2.
-The standard deviation of X is √(1- p) / p2
The probability that it takes more than n trials to see the first success is
P(X > n) = (1 – p)n
*Geometric distributions are always skewed RIGHT.