Bernoulli distributions and probability models
A Bernoulli trial is where an event happens and there are only two possible
outcomes (success or failure) common examples of a Bernoulli trial are
flipping a coin, shooting free throws, checking for defective products
coming off an assembly line . . . .
Conditions for the
Bernoulli model:
1) two possible
outcomes
2) each outcome is
independent from
one another
3) the probability of
success remains
constant
The two types of Bernoulli trials we will study are called
1) geometric trials
This is where we are trying to figure out how many trials need to be done before we
get our first success (the probability and the expected number of trials)
Let p = probability of a success and q = probability of failure
X = number of trials until first success
THE PROBABILITY OF X = SOME NUMBER P(X=x) = qx-1p
1
The expected value of X: E(X) =
p
The standard deviation of X =
q
p2
2) binomial trials
This is where we are trying to figure out the probability of obtaining a specific
number of successes in a set amount of trials (or the expected number of successes in
a set number of trials)
Let p = probability of a success and q = probability of failure and n = the number of trials
X = number of successes in n trials
THE PROBABILITY OF X = EXACTLY SOME NUMBER P(X=x) = (nCx)(px)(qn-x)
The expected value of X: E(X) = np
The standard deviation of X = npq
Binompdf(n,p,X) will calculate the probability of getting exactly X successes in n trials if
each success has a p chance of happening
Binomcdf(n,p,X) will calculate the probability of getting at most X successes in n trials if
each success has a p chance of happening
1- Binomcdf(n,p,X-1) will calculate the probability of getting at least X successes in n
trials if each success has a p chance of happening
If we need to do a binomial probability and n is really large we once again rely on
THE NORMAL MODEL (that’s why it’s nice to have the expected value (mean) and the
standard deviation)
A binomial probability distribution works pretty well when np and nq are at least 10
(That’s the only condition to check if you are going to approximate a binomial
distribution with THE NORMAL MODEL)
EXAMPLES:
1) a regular golfer has a 15% chance of getting a birdie on any hole
a) What is the probability the golfer will get his first birdie on the 5th hole he plays?
b) What is the probability the golfer will get exactly 5 birdies in 18 holes?
c) What is the probability the golfer will get at most 5 birdies in 18 holes?
d) What is the probability the golfer will get at least 5 birdies in 18 holes?
e) What is the probability the golfer will get at least 15 birdies in 72 holes?
2) An Olympic archer is able to hit the bull’s eye of a target 80% of the time.
Assume each shot is independent of the others. If she shoots 6 arrows, what is the
probability of . .
a. Her first bull’s eye happens on her third shot
b. She misses the bull’s eye at least once
c. Her first bull’s eye happens on the fourth or fifth shot
d. She gets exactly 4 bull’s eyes
e. She gets at least 4 bull’s eyes
f. She gets at most 4 bull’s eyes
g. Suppose now she shoots 200 arrows
i. What are the mean and standard deviation for the number of bull’s
eyes she might get?
ii. Could a normal model be appropriate in this case?
iii. What is the probability she gets more than 190 bull’s eyes?
iv. Would you be surprised if she only made 140 bull’s eyes?