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Geometric and Hyper-geometric Distribution
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Take a fair coin and toss it as many times as
needed until you observe a head.
Let X= number of tosses that is needed.
Sample points={H, TH, TTH, TTTH, …}
Distribution of X
X
1 (H)
2(TH)
3(TTH) 4(TTTH) K(TTT…TH)
P(X)
1/2
1/4
1/8
1/16
?
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X
Think about another example, if we keep tossing
a biased coin with 70% of getting a tail and 30% of
seeing a head, let X=number of tosses needed to
get the first head.
Then the distribution of X is:
1 (H) 2(TH)
P(X) 0.3
0.7*0.3
3(TTH)
4(TTTH)
K(TTT…TH)
0.7*0.7*0.3
0.7*0.7*0.7*0.3
0.7*0.7*…*0.7*0.3
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If we repeat an experiment with two
outcomes, success/failure, with probability p
for success and q=1-p for failure, the number
of trials needed to get the first success
follows a Geometric distribution, say,
X~Geo(p).
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Similarities and differences between Binomial
and Geometric random variable.
◦ Similarities: independent trials of the identical
experiment, probabilities of success/failure
consistent.
◦ Differences:
 1. For Binomial, we know how many trials we have in
total and for Geometric, we don’t know it, actually that
number is not of interest.
 2. For Binomial, there are usually several possibilities
for a specific value of the variable, but for geometric,
there is only one. (note: there is a coefficient of a
combination for the binomial probability)
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Someone is trying to take the road test to get
a driver’s license. If the probability of passing
the test is 40%, what is the probability that
this person will pass the test at second shot?
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What is the probability that someone will pass
the road test in 5 trials?
Given that someone has taken the test 4
times and still has not got the license, what is
that person’s chance of passing it the next
time?
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X~Geo(p), then
◦ E(X)=1/p
◦ Var(X)=(1-p)/(p^2)
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On average, how many times does one have
to take the test to get the driver’s license?
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In one of the episodes of Planet Earth by BBS,
an experiment was recorded. In the
experiment, some animal scientists captured
a snow leopard and put an instrument with a
remote sensor so that they can keep track of
her behavior. Also, they can use the
instrument to estimate the number of wild
snow leopard existing.
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Suppose there are a total of N wild snow
leopards, r of them are captured and attached
the instrument and the rest N-r are not.
In the future, scientists can keep capturing
wild snow leopards and count how many
captured ones have the instrument and how
many don’t. (assuming snow leopards don’t
take the instrument off by themselves).
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In a population of size N, r have a feature of
interest and N-r don’t. Then if we take a
sample of size n from the population, what is
the probability that x units in our sample of
size n have that feature?
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Hyper-geometric distribution:
If a random variable follows a hypergeometric distribution, we say X~HG(N, n, r)
and
◦ p(x=k)=
◦ Parameters:
 N: number of elements in the population
 n: number of elements in the sample
 r: number of elements in the population with the
desired feature.
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Population of size N, r with desired feature.
◦ If we take samples, considered a trial, with
replacement, then each trial is independent with the
same probability of getting one with the desired
feature. (Binomial)
◦ If we take samples without replacement, then each
trial is not independent. Also, the probability of
getting one with the desired feature changes across
trials. (Hypergeometric)
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If we have 20 books, 4 textbooks and 16
non-textbooks.
A. If we take one book at a time and then put
it back, repeat 5 times, what is the probability
that we get all 4 textbooks?
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B. If we take one book at a time and do not
put it back, repeat 5 times, what is the
probability that we get all 4 textbooks?
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C. If we take 5 books out of the 20, what is
the probability that we get all 4 textbooks?
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Given that a random variable follows a Hypergeometric distribution,
◦ E(X)=n*r/N
◦ Var(X)=n(r/N)(1-r/N)[(N-n)/(N-1)]
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D. On average, if we take 5 books out of the
20, how many textbooks can we get?
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A. Suppose we randomly draw 5 cards from a
deck of 52 cards, what is the probability that
there are exactly 2 red cards in our hand?
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B. what is the probability that we get at most
two hearts?
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Four players are playing a poker game out of
a deck of 52 cards. Each player has 13 cards.
Let X be the number of Kings one player may
have, and answer the following questions.
1. Is X a discrete or continuous random
variable?
2. Find an appropriate probability distribution
that can be used to describe X. Also, find the
corresponding parameter(s).
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3. Find the sample space and X and the
probability corresponding to each point in the
sample space.