Independent Events
Discrete Math - Section 7.5
I. Warm-Up Problem
Scenario A: One card is chosen from a standard deck of 52 cards. Define events as follows:
• J: "The card is a jack"
• F: "The card is a face card."
Determine the following probabilities:
a.
P(J)
d.
P(J | F)
b.
P(F)
e.
P(F | J)
c.
P(J ∩ F)
f.
P(F) P(J)
Then answer these questions:
g. Does knowing that the card is a face card improve the chances of determining if the card is a jack? Explain.
h.
Does knowing that the card is a jack improve the chances of determining if the card is a face card? Explain.
Scenario B: One card is chosen from a standard deck of 52 cards. Define events as follows:
• R: "The card is red"
• Q: "The card is a queen."
Determine the following probabilities:
i.
P(R)
l.
P(R | Q)
j.
P(Q)
m. P(Q | R)
k.
P(R ∩ Q)
n.
P(R) P(Q)
Then answer these questions:
o. Does knowing that the card is a queen improve the chances of determining if the card is red? Explain.
p.
Does knowing that the card is red improve the chances of determining if the card is a queen? Explain.
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II. Formalization of Independent Events
We now use our knowledge of conditional probability to determine whether or not events are independent.
Intuition:
Two events are independent if one does not affect the other
Formal Definition:
Let E and F be two events of a sample space S with P(F) > 0. We say event E is independent of event F iff P(E|F) = P(E).
Theorem:
Let E and F be independent events such that P(E) > 0 and P(F) > 0. If E is independent of F, then F is independent of E.
Formal Definition:
If two events E and F have positive probabilities and if E is independent of F, then F is independent of E. In this case, we say
E and F are independent events.
Test for Independence:
Given two events E and F of sample space S,
E, F are independent events
⇔ P(E ∩ F) = P(E) P(F)
So… if we are asked to prove two events independent, we have two options:
• Use the first formal definition and prove one event independent of the other. Then either appeal to the theorem to
claim the independence goes both ways OR use the formal definition in the other direction.
• Use the test for independence.
Example 1: Eye Color
Suppose 36 students are surveyed. We have the following events:
E: student has blue eyes
F: student is female
We are given the following data table:
Blue eyes
Not blue eyes
Totals
Female
12
12
24
Not Female
6
6
12
Totals
18
18
36
Are E and F independent events?
a.
Prove or disprove using the definition.
b.
Prove or disprove using the test for independence.
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Independence for More Than Two Events
A set {E1, E2, …, En} of n events is called independent if the occurrence of one or more of the events does not change the
probability of any of the others. Symbolically,
P(E1 " E2 "K " En ) = P(E1) # P(E2 ) #K # P(En )
Example 2: Skin Cream
A new skin cream can cure infection 90% of the time. If 5 randomly selected people with skin infections use this cream,
! is the probability that…
assuming independence, what
a. …all five are cured?
b.
…all five still have the infection?
Practice Problem
If P(A) = 0.8, P(C) = 0.95 and P(A ∩ C) = 0.76, are events A and C independent?
III. More Practice
Problem 1
If E and F are independent events, P(E) = 0.3, and P(E ∩ F) = 0.18, find…
a. P(F)
b.
P(E|F)
c.
P(F|E)
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Problem 2
Let P(E) = 0.8 and P(F) = 0.6. If E and F are independent events, find the following:
a. P(E ∩ F)
b.
P(E ∪ F)
c.
P(E|F)
d.
P(F|E)
Problem 3
If E and F are independent events, find P(E) if P(F) = 0.4 and P(E ∪ F) = 0.82.
Problem 4
Events A, B, C, and D are independent events with P(A) = 0.6, P(B) = 0.3, P(C) = 0.5, and P(D) = 0.4.
Determine P(A ∩ B ∩ C ∩ D).
Problem 5
If P(E) = 0.8, P(F) = 0.3, and P(E ∪ F) = 0.82, find P(E ∩ F). Are E and F independent events? Justify your answer.
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IV. Wrapping Up Multiplication Rules
From our discussion in these past two sections we have the following multiplication rules.
Multiplication Rules
(1) If two events are independent, then P(A ∩ B) = P(A) P(B)
(2) If two events are dependent, then P(A ∩ B) = P(A) P(B|A)
Example (Freund)
In a fifth grade class of 18 boys and 12 girls, one pupil is chosen each week by lottery to act as an assistant to the teacher.
What is the probability that a girl will be chosen two weeks in a row if
a.
the same pupil cannot serve two weeks in a row?
b.
the restriction of part (a) is removed?
Note of importance: If A and B are mutually exclusive events then P(A ∩ B) = ___
Homework: 7.5. #1-13 odd, 31-32
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