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Activity 6 - 4
Conditional Probabilities
German research vessel POLARSTERN weather station, 2007
Objectives
• Identify a conditional probability problem
• Determine conditional probabilities using sample
space or the data from a table
• Determine conditional probabilities using a formula
Vocabulary
• None new
Activity
It has rained for the past three days. However, the
weather forecast for tomorrow’s softball game is sunny
and no chance of rain. You decide not to take a rain
jacket, only to get drenched when there is a sudden
downpour during the game.
Although not perfect, the level of accuracy of weather
forecasting has increased significantly through the use of
computer models. These models analyze current data and
predict atmospheric conditions at some short period of
time form that moment. Based on these predictions,
another set of atmospheric conditions are predicted. This
process continues until the forecast for the day and the
extended forecast for the next several days are
completed.
Activity cont
What does the level of accuracy in the weather forecasting
model described above depend upon?
The level of accuracy decreases as the forecasts extend
several days ahead. Explain.
The accuracy of weather forecasting is based on
conditional probabilities; that is, the probability that one
event happens, given that another event has already
occurred. For example, the probability that it will rain this
afternoon, given that a low pressure system moved into
the area this morning, is a conditional probability.
Conditional Probabilities
Given that some event has already occurred,
what is the probability that another event could
occur?
• P(A | B) is read what is the probability of A
given that B has occurred
• It is governed by the following formula:
P( A and B)
P(A | B) = ------------------P(B)
obviously P(B) ≠ 0 (since it has occurred)
Example 1
Given the following information: P(A) = 0.75,
P(B) = 0.6 and P(A and B) = 0.33
What is the P(A | B)?
P( A and B)
0.33
P(A | B) = ------------------- = ---------- = 0.556
P(B)
0.60
What is the P(B | A)?
P( A and B)
0.33
P(B | A) = ------------------- = ---------- = 0.444
P(A)
0.75
Contingency Tables
Given information about the number of occurrences of
events in tables (experimental probability), we can
calculate probabilities (including conditional
probabilities) using the following formula:
n( A and B)
P(A | B) = ------------------n(B)
where
n(A and B) represents the number of observations of
both A and B; and
n(B) represents the total number of observations of B
Contingency Tables Example 1
Right handed
Left handed
Total
Male
48
Female
42
Total
90
12
60
8
50
20
110
1. What is the probability of left-handed given that
it is a male? P(LH | M) = 12/60 = 0.20
2. What is the probability of female given that they
were right-handed?
P(F| RH) = 42/90 = 0.467
3. What is the probability of being left-handed?
P(LH) = 20/110 = 0.182
Contingency Tables Example 2
Involved in Sports
Not involved in Sports
Total
Male Female Total
52
58
110
32
84
58
116
90
200
1. What is the probability of being involved in sports
given that it is a male?
P(S | M) = 52/84 = 0.619
2. What is the probability of female given that they were
involved in sports?
P(F | S) = 58/110 = 0.527
3. What is the probability of not being involved in sports?
P(NS) = 90/200 = 0.45
Summary and Homework
• Summary
– Conditional probability, P(A|B), is read probability
of A, given B has already occurred
– For contingency tables, probabilities are the
number of occurrences listed in the table
– Conditional probability formula:
P(A and B)
P(A|B) = ----------------P(B)
• Homework
– pg 743 – 747; problems 1-3, 5, 8