ACCA F5 - Performance Management
Joint Probability: Joint
probability represents the chance of occurring an event for other estimate /
expectation. Joint probability is the probability that two events will occur
simultaneously.
Here, we have a table with
joint probability. For joint probability, both rows and columns represent
events.
EVENT 1
\ EVENT 2
|
Outcome Cloudy
|
Outcome Sunny
|
Forecast Cloudy
|
16%
|
2%
|
Forecast Sunny
|
4%
|
78%
|
Marginal probability:
Marginal probability is the probability of occurrence of the single event.
EVENT 1
\ EVENT 2
|
Outcome Cloudy
|
Outcome Sunny
|
Marginal probability for forecast
|
Forecast Cloudy
|
16%
|
2%
|
18%
|
Forecast Sunny
|
4%
|
78%
|
82%
|
Marginal probability for outcome
|
20%
|
80%
|
100%
|
Conditional probability:
Probability of occurrence of one event given the other event already occurred.
Here in this example we assume calculate the probability outcome for the
forecast made.
EVENT 1
\ EVENT 2
|
Outcome Cloudy
|
Outcome Sunny
|
Total
|
Forecast Cloudy
|
16% / 18% = 88.9%
|
2% / 18% = 11.1%
|
100%
|
Forecast Sunny
|
4% / 82% = 4.88 %
|
78% / 82% = 95.12%
|
100%
|
Value of imperfect
information: Imperfect information accounts for uncertainty. Uncertainty is
exposed by the condition imposed to information accuracy. Only after an event
happens one can predict likely outcome. What if somebody makes a market survey
before launching a product? Can the market surveyor accurately predict likely
future sales for a product? For above example think, how precise weather
forecast can be made for a particular day. Past analysis of forecast forms
reasonable condition on the accuracy level of future forecast.
Calculation of expected
values for imperfect information (forecast)
Method 1 Using Joint
probability:
Step 1
Forecast
|
Outcome
|
Cloudy
|
Sunny
|
Cloudy
|
Medium Batch
|
337.5
|
600
|
Sunny
|
Large Batch
|
80
|
1180
|
Step 2
Forecast
|
Outcome
|
Cloudy
|
Sunny
|
Total
|
Cloudy
|
Medium Batch
|
337.5*16%=54
|
600*2%=12
|
=54+12=56
|
Sunny
|
Large Batch
|
80*4%=3.2
|
1180*78%=920.4
|
=920.4+3.2=923.6
|
Total EV
|
=56+923.6=989.6
|
Method 2, Using conditional
probability and marginal probability
Now, maximizing payoff in
line with imperfect information
Step 1
Forecast
|
Outcome
|
Cloudy
|
Sunny
|
Cloudy
|
Medium Batch
|
337.5
|
600
|
Sunny
|
Large Batch
|
80
|
1180
|
Calculation of expected
values for imperfect information (forecast)
Step 2
Forecast
|
Outcome
|
Cloudy
|
Sunny
|
Total EV for forecast
|
Cloudy
|
Medium Batch
|
337.5*88.9%=300
|
600*11.1%=66.6
|
300+66.6=366.6
|
Sunny
|
Large Batch
|
80*4.88%=3.9
|
1180*95.12%=1122.4
|
3.9+1122.4=1126.4
|
Step 3
Forecast
|
Marginal probability for forecast
|
Total EV for forecast
|
EV for imperfect information
|
Cloudy
|
18%
|
366.6
|
18%*366.6=66
|
Sunny
|
82%
|
1126.4
|
82%*1126.4=923.6
|
Total expected value
|
=66+923.6=989.6
|
EV's for imperfect
information: Extra payoff generated under imperfect information to that of best
alternative.
EV's for imperfect
forecast
|
Expected value forecast -
Expected value for best alternative
= 989.6 - 960
= 29.6
|
The Tree
Joint Probability: Joint
probability represents the chance of occurring an event for other estimate /
expectation. Joint probability is the probability that two events will occur
simultaneously.
Here, we have a tree with
joint probability. For joint probability, both rows and columns represent
events.
Prior Event
|
Event 1
|
Posterior Event
|
Event 2
|
Joint probability
|
Forecast Cloudy
|
Outcome Cloudy
|
16%
|
||
Outcome Sunny
|
2%
|
|||
Forecast Sunny
|
Outcome Cloudy
|
4%
|
||
Outcome Sunny
|
78%
|
Marginal probability: Marginal
probability is the probability of occurrence of the single event.
Event 1
|
Event 2
|
Joint probability
|
Forecast Cloudy
|
Forecast Sunny
|
Outcome Cloudy
|
Outcome Sunny
|
Forecast Cloudy
|
Outcome Cloudy
|
16%
|
16%
|
16%
|
||
Outcome Sunny
|
2%
|
2%
|
2%
|
|||
Forecast Sunny
|
Outcome Cloudy
|
4%
|
4%
|
4%
|
||
Outcome Sunny
|
78%
|
78%
|
78%
|
|||
Marginal Probability
|
18%
|
82%
|
20%
|
80%
|
||
Conditional probability:
Probability of occurrence of one event given the other event already occurred.
Here in this example, we calculate the probability outcome for the forecast
made.
In mathematical term, joint
probability divided by marginal probability is conditional probability.
Given that forecast is already made, the condational probability is
|
Event 1
|
Event 2
|
Joint probability
|
Total
|
Forecast Cloudy
|
Outcome Cloudy
|
16% / 18% = 88.9%
|
= 88.9% +11.1%
=100%
|
|
Outcome Sunny
|
2% / 18% = 11.1%
|
|||
Forecast Sunny
|
Outcome Cloudy
|
4% / 82% = 4.88 %
|
=4.88% + 95.12%
=100%
|
|
Outcome Sunny
|
78% / 82% = 95.12%
|
Value of imperfect
information: Imperfect information accounts for uncertainty. Uncertainty is
exposed by the condition imposed to information accuracy. Only after an event
happens one can predict likely outcome. What if somebody makes a market survey
before launching a product? Can the market surveyor accurately predict likely
future sales for a product? For above example think, how precise weather
forecast can be made for a particular day. Past analysis of forecast forms
reasonable condition on the accuracy level of future forecast.
Calculation of expected
values for imperfect information (forecast)
Method 1 Using Joint
probability:
Step 1
Given that forecast is already made, the choice depends on forecast
|
Event 1
|
Choice
|
Event 2
|
EV's
|
Forecast Cloudy
|
Medium batch
|
Outcome Cloudy
|
337.5
|
|
Outcome Sunny
|
600
|
|||
Forecast Sunny
|
Large batch
|
Outcome Cloudy
|
80
|
|
Outcome Sunny
|
1180
|
Step 2
Event 1
|
Choice
|
Event 2
|
EV's
|
Total
|
Forecast Cloudy
|
Medium batch
|
Outcome Cloudy
|
337.5*16%=54
|
=54+12=56
|
Outcome Sunny
|
600*2%=12
|
|||
Forecast Sunny
|
Large batch
|
Outcome Cloudy
|
80*4%=3.2
|
=920.4+3.2=923.6
|
Outcome Sunny
|
1180*78%=920.4
|
|||
Total EV
|
=56+923.6=989.6
|
|||
EV's for imperfect
information: Extra payoff generated under imperfect information to that of best
alternative.
EV's for imperfect
forecast
|
Expected value forecast -
Expected value for best alternative
= 989.6 - 960
= 29.6
|
Method 2, Using conditional
probability and marginal probability
Now, maximizing payoff in
line with imperfect information
Step 1
Given that forecast is already made, the choice depends on forecast
|
Event 1
|
Choice
|
Event 2
|
EV's
|
Forecast Cloudy
|
Medium batch
|
Outcome Cloudy
|
337.5
|
|
Outcome Sunny
|
600
|
|||
Forecast Sunny
|
Large batch
|
Outcome Cloudy
|
80
|
|
Outcome Sunny
|
1180
|
Calculation of expected values
for imperfect information (forecast)
Step 2
Event 1
|
Choice
|
Event 2
|
EV's
|
Total
|
Forecast Cloudy
|
Medium batch
|
Outcome Cloudy
|
337.5*88.9%=300
|
300+66.6=366.6
|
Outcome Sunny
|
600*11.1%=66.6
|
|||
Forecast Sunny
|
Large batch
|
Outcome Cloudy
|
80*4.88%=3.9
|
3.9+1122.4=1126.4
|
Outcome Sunny
|
1180*95.12%=1122.4
|
Step 3
Forecast
|
Marginal probability for forecast
|
Total EV for forecast
|
EV for imperfect information
|
Cloudy
|
18%
|
366.6
|
18%*366.6=66
|
Sunny
|
82%
|
1126.4
|
82%*1126.4=923.6
|
Total expected value
|
=66+923.6=989.6
|
EV's for imperfect
information: Extra payoff generated under imperfect information to that of best
alternative.
EV's for imperfect
forecast
|
Expected value forecast -
Expected value for best alternative
= 989.6 - 960
= 29.6
|
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