Payoff Table and Tree Part4

ACCA F5 - Performance Management 

Bayes' Theorem: It demonstrates the interrelation between joint, marginal and conditional probability. Bayes' theorem is used to predict the conditional cross probability for one joint event given conditional cross probability for other joint event.

Joint probability

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 for given forecast

CONDITION \ EVENT 2	Marginal probability	Outcome Cloudy	Outcome Sunny	Total
Forecast Cloudy	18%	16% / 18% = 88.9%	2% / 18% = 11.1%	100%
Forecast Sunny	82%	4% / 82% = 4.88 %	78% / 82% = 95.12%	100%
Total		=88.9% + 4.88% = 93.78%	=11.1% + 95.12% = 106.22	=200%

Straight way for alternate conditional probability

Conditional probability when joint probabilities are known,

EVENT 1      \    EVENT 2	Outcome Cloudy	Outcome Sunny	Total
Marginal probability	20%	80%
Forecast Cloudy	16% / 20% = 80%	2% / 80% = 2.5%	80% + 1.6% = 82.5%
Forecast Sunny	4% / 20% = 20%	78% / 80% = 97.5%	20% + 97.5% = 117.5%
Total	100%	100%	200%

Revised Bayes' Theory for known conditional probability for given forecast revised to give joint probability

EVENT 1      \    EVENT 2	Outcome Cloudy	Outcome Sunny	Total
Forecast Cloudy	88.9%*18%= 16%	11.1%*18%= 2%	18%
Forecast Sunny	4.88%*82%= 4%	95.12%*82%= 78%	82%
Total	20%	80%	100%

Once joint probability is known, follow above step again.

EVENT 1 \  CONDITION	Outcome Cloudy	Outcome Sunny	Total
Marginal probability	20%	80%
Forecast Cloudy	16% / 20% = 80%	2% / 80% = 2.5%	80% + 1.6% = 82.5%
Forecast Sunny	4% / 20% = 20%	78% / 80% = 97.5%	20% + 97.5% = 117.5%
Total	100%	100%	200%

The tree

Bayes' Theorem: It demonstrates the interrelation between joint, marginal and conditional probability. Bayes' theorem is used to predict the conditional cross probability for one joint event given conditional cross probability for other joint event.

Joint probability

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%

Straight way for alternate conditional probability

Conditional probability when joint probabilities are known,

Given that outcome is known, the condition for forecast is	Event 1	Event 2	Marginal Probability	Joint probability	Total
	Forecast Cloudy	Outcome Cloudy	20%	16% / 20% = 80%	80% + 1.6% = 82.5%
		Outcome  Sunny	80%	2% / 80% = 2.5%
	Forecast Sunny	Outcome Cloudy	20%	4% / 20% = 20%	20% + 97.5% = 117.5%
		Outcome  Sunny	80%	78% / 80% = 97.5%

Conditional probability for given forecast

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%

Revised Bayes' Theory for known conditional probability for given forecast revised to give joint probability

Event 1	Event 2	Joint probability	Outcome cloudy	Outcome sunny
Forecast Cloudy	Outcome Cloudy	88.9%*18%= 16%	16%
	Outcome  Sunny	11.1%*18%= 2%		2%
Forecast Sunny	Outcome Cloudy	4.88%*82%= 4%	4%
	Outcome  Sunny	95.12%*82%= 78%		78%
Marginal Probability			20%	80%

Once joint probability is known, iterate the calculation step shown in:

Straight way for alternate conditional probability

Decision Making

Joint, Marginal, and Conditional Probability

Bayes Theorem