Unit 7
Probability Distributions
Day 4
Hypergeometric Distributions
Hypergeometric Distribution
• Consider a baseball coach choosing the
starting line-up for a game
• The same person cannot be chosen twice
• Once a person is chosen it reduces the
number of possibilities for the next
position
• The trials in this situation are dependent
• Hypergeometric distribution
• Involves a series of dependent trials
• Each trial has two outcomes, success or
failure
• The probability of success changes with
each trial
• The random variable, 𝑋, is the number of
successful trials in the experiment
Example 1
Determine the probability distribution for the
number of women on a 4 person committee to be
chosen from 5 women and 7 men.
Let 𝑋 represent the number women on the
committee
𝑋
0
1
2
3
4
𝑃(𝑋 = 𝑥)
Probability for a Hypergeometric
Distribution
𝑎 𝑛−𝑎
𝑃 𝑋=𝑥 =
𝑥
𝑟−𝑥
𝑛
𝑟
where 𝑎 is the number of successful
outcomes possible and 𝑥 the number of
successes wanted
𝑛 is the total number of possible
outcomes
𝑟 is the total number of trials
Example 2
Joe deals 5 cards to Sally. What is the probability
that she has at least 4 spades in her hand?
Let 𝑋 rep the number spades
𝑎 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘
𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒𝑠 𝑖𝑛 ℎ𝑎𝑛𝑑
𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 𝑑𝑒𝑎𝑙𝑡
𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑃 𝑋 ≥ 4 = 𝑃 𝑋 = 4 + 𝑃(𝑋 = 5)
=
13𝐶4
× 39𝐶1 13𝐶5
+
52𝐶5
52𝐶5
= 0.107 + 0.0005
= 0.1075
Expected Value for a Geometric Distribution
𝑟𝑎
𝐸 𝑋 =
𝑛
Example 3
What is the expected number of spades in Sally’s
hand?
Let 𝑋 rep the number spades
𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
𝑎 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘
𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠
𝑟𝑎 5 × 13
𝐸 𝑥 =
=
= 1.25 𝑠𝑝𝑎𝑑𝑒𝑠
𝑛
52
You would expect to have 1.25 spades in the hand.
Example 4
A jar contains 20 red and 30 green jellybeans.
a. What is the probability that at least two are red
when you draw 10 jellybeans from the jar?
Let 𝑋 represent the number of red jelly beans.
𝑎 = 20
𝑟 = 10
𝑛 = 50
𝑃 𝑋 ≥ 2 = 1 − 𝑃 𝑋 = 0 − 𝑃(𝑋 = 1)
=1−
20𝐶0
× 30𝐶10 20𝐶1 × 30𝐶9
−
50𝐶10
50𝐶10
= 1 − 0.003 − 0.028
= 0.969
b. What is the expected number of red jelly beans?
𝑟𝑎
𝐸 𝑥 =
𝑛
10(20)
𝐸 𝑥 =
50
𝐸 𝑥 =4
The expected number of red jelly beans is 4.
Suggested Exercises
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Page 404 # 1, 2a, 3, 7, 8, 9, 10, 12