Use Bayes' Theorem or a Tree Diagram to Calculate the Indicated

question 244

Short Answer

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places.
​ $Y _ { 1 }$ , $Y _ { 2 }$ , $Y _ { 3 }$ form a partition of S.
​ $P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.4$ , $P \left( X \mid Y _ { 3 } \right) = 0.5$ , $P \left( Y _ { 1 } \right) = 0.2$ , $P \left( Y _ { 2 } \right) = 0.5$ .
​
Find $P \left( Y _ { 2 } \mid X \right)$ .
​ $P \left( Y _ { 2 } \mid X \right) =$ __________

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Definitions:

Limited Liability Firm

A type of business entity where the owners' liability for debts is restricted to the amount they invested in the company, protecting personal assets from business liabilities.

Sole Proprietorship

A business owned and operated by a single individual, where there is no legal distinction between the owner and the business entity.

Communist

A political and economic ideology advocating for a classless society, where property and resources are communally owned.

Socialist

Pertaining to a political and economic theory advocating for the means of production, distribution, and exchange to be owned or regulated by the community as a whole.