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Independence and Tree Diagrams
Slideshow 56, Mathematics
Mr Richard Sasaki, Room 307
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Objectives
• Review multiplying decimal numbers less than 1
• Learn some new notation about different events
• Recall how to make calculations with independent events
• Introduce Tree Diagrams
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Decimal Numbers
We will start using decimal numbers in probability (as well as fractions).
Let’s have a bit of practice multiplying and dividing some!
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0.03 0.18
0.21 0.42
0.021 0.092
0.0045 0.07
0.5 0.3
0.9 0.3
0.125 0.08
0.01 0.375
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Notation
Do you remember how to calculate the probability of two independent events occurring successfully?ExampleWe roll an unbiased regular die and a spin an unbiased spinner with values 1, 2 and 3 in that order.
What is the probability of getting a 5 and then a 2?
P(5, 2) = 16×13
¿118
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Notation
When independent events take place, we multiply both of their probabilities to find the probability of them both occurring.
P(AB) = P(A) P(B)Note – This is only true when events A and B are independent!Also, the complement of A is written A’.
P(A’) P(A)
Do you remember how to write “the probability of event A and B”?
P(AB)
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P(A’) = 0.6
P() = 0.4 )
P() = 0.4 )
P() = 0 because A can’t happen and not happen at the same time.)
Replacing / with replacement)
P(A) = P(Jack) =
P(B) = P(Red) =
P(A’) = 1 – P(A) =
P() )
BecauseP(B) = P(B’) = ½
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Tree Diagrams
We can represent different possible outcomes with tree diagrams. Branches represent independent events. Let’s consider two events with only success and fail outcomes.
Event A Event B
P(A)
P(A’)
P(B)P(B’)
P(B)P(B’)
P(AB)
P(AB’)P(A’B)
P(A’B’)
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Example
Event A Event B
P(A)
P(A’)
P(B)
P(B’)
P(B)
P(B’)
P(AB)
P(AB’)P(A’B)
P(A’B’)
Two independent events A and B occur in order where P(A) = 0.3 and P(B) = 0.9. Represent all outcomes with a tree diagram and calculate P(AB’) and P(A’B’).
0.3
0.7
0.9
0.9
0.1
0.1
=0.03
=0.07
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Answers - Easy
P()
P()
P()
P()
P()P()
12
12
1212
1212
P() )P() )
P() P()
P()
P()
P()
P()
P()
P()
0.25
0.75
0.2
0.20.8
0.8
P() )
P() )
P(AB) = P(A) P(B).BecauseSo A and B must be independent.
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Answers - Hard)
)
))
A B
1747
27
27
27
27
37
37
37
27
27
27
1000)
100)
P(Other)
1000)
1000)
1000)
100)
100)
100)
P(Other)
P(Other)
)
P(Two ¥ 100) =
P(¥ 1000 and ¥ 100) =
P(At least one ¥ 1000) =
P(No ¥ 100 or ¥ 1000) =