independence and tree diagrams slideshow 56, mathematics mr richard sasaki, room 307
TRANSCRIPT
Independence and Tree Diagrams
Slideshow 56, Mathematics
Mr Richard Sasaki, Room 307
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
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!
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
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
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)
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
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’)
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
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.
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) =