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

question 131

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.
​ $P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.2$ , $P \left( X \mid Y _ { 3 } \right) = 0.6$ , $P \left( Y _ { 1 } \right) = 0.3$ , $P \left( Y _ { 2 } \right) = 0.4$ .
​
Find $P \left( Y _ { 1 } \mid X \right)$ .
​ $P \left( Y _ { 1 } \mid X \right) =$ __________

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Definitions:

Least Squares Regression

A statistical method for estimating the relationship between variables by minimizing the squares of the differences between observed and predicted values.

Relationship

A connection, association, or involvement between two or more variables or entities.

Standard Error

The standard deviation of the sampling distribution of a statistic, most commonly the mean.

Regression Slope

Indicates the steepness and direction of the linear relationship between two variables on a regression plot.