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QUESTIONS:
2014; 3d
2013; 1a
2013; 1b
probability trees
A
=
x B
A’
A.B = AB
B’
B
B’
A
+
A’
B+
B’
B
B’
A+A’ = 1 AB+AB’ = A
Probability trees show a sequence of
two or more events and the probabilities assciated with these events
A
A’
AB+AB’+A’B+A’B’ = 1
for instance
Each individual set of branches, (associated with an
event and it’s comlementary event(s)) has a total probability of 1. For example the sum of the event A, and its
complementary event A’, is 1.
Multiply along the branches to get the probability of the
intersection of events. For example, A.B= AB.
Add between the branches to get the probability of the
union of events.
B+
B’
+
B
+
B’
A research scientist plants plants 11 seeds in
Plot A and 9 seeds in Plot B. The probability that
a seed germinates in Plot A is 0.7 and in Plot B is
0.8.
Find the probability that a seed chosen at random germinates.
All probabilities possible from the intersection of events , indicated at the far right of
the probability tree, will equal a sum of 1.
For example, the intersection of events A
or A’ and B or B’ give the proabilities of AB,
AB’, A’B and A’B’ will add together to give a
sum of 1.
The sum of the probability of the intersection of events occuring, is equal to the probability of the event that happened initially.
For example, AB+AB’ =A.
practice Question
Solve
A research scientist plants plants 11 seeds in Plot A and 9 seeds in Plot B.
The probability that a seed germinates in Plot A is 0.7 and in Plot B is 0.8.
Find the probability that a seed chosen at random germinates.
Step One
Step Two
Establish variables and their
relationship and form tree
Find the required probabilities
Variables
variable 1: Plots (A or B)
variable 2: Germinate (germinate or not)
GERMINATE
Form probability
tree
PLOT A
Calculate probabilities of the
intersection of
events
9/20
0.3
NOT
= 0.165
GERMINATE
PLOT A
GERMINATE = 0.27
PLOT B
NOT
GERMINATE = 0.18
0.4
GERMINATE
11/20
GERMINATE = 0.385
0.6
9/20
NOT
GERMINATE
PLOT B
Add probabilities
11/20
0.7
P(PLOT A GERMINATE)= PLOT A . GERMINATE
= 11/20 . 0.7 = 0.385
NOT
GERMINATE
P(PLOT A NOT GERMINATE)= PLOT A . NOT GERMINATE
= 11/20 . 0.3 = 0.165
0.7
GERMINATE
P(PLOT B GERMINATE)= PLOT B . GERMINATE
= 9/20 . 0.6 = 0.27
0.3
NOT
GERMINATE
P(PLOT B NOT GERMINATE)= PLOT B . NOT GERMINATE
= 9/20 . 0.4 = 0.18
0.6
GERMINATE
0.4
NOT
GERMINATE
PLOT A
PLOT B
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Probability that a
seed chosen at random germinates
P(SEED GERMINATES)= 0.385 + 0.27
= 0.655
= 0.655
0.7
11/20
PLOT A
0.3
0.6
9/20
PLOT B
0.4
GERMINATE = 0.385
NOT
= 0.165
GERMINATE
GERMINATE = 0.27
NOT
GERMINATE = 0.18