The problem reads: Suppose P(X_1)=.75 and P(Y_2|X_1)=.40. What is the joint probability of X_1 and Y_2? This is how I answered it. P(X_1 and Y_2) =P(X

Khaleesi Herbert

Khaleesi Herbert

Answered question

2021-01-10

The problem reads: Suppose P(X1)=.75 and P(Y2X1)=.40. What is the joint probability of X1 and Y2?
This is how I answered it. P(X1 and Y2) =P(X1)×P(Y1X1)=.75×.40=0.3.
What I don't understand is how do you get the P(Y1X1)? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?

Answer & Explanation

Faiza Fuller

Faiza Fuller

Skilled2021-01-11Added 108 answers

X1 and Y2 both represent the occurrence of someevent, e.g. the number that comes up after rolling a fair die or if a person is a smoker or a non-smoker. P(X1) represents the probability that the event X1 occurs. P(Y2X1) represents the probability that the event Y2 occurs given that the event X1 has occurred. In other words, we wantto know what chance Y2 will occur knowing that X1 has already happened.
The joint probability of X1 and Y2, denoted by P(X1Y2) or P(Y2X1)
P(X1Y2)=P(Y2X1)
by the commutative property, representsthe probability that both events X1 and Y2 occur.
Use the following identity to compute
P(Y2X1)=P(X1Y2)P(X1)
The formula for P(Y2X1) can be derived in the following manner: If the event X1 occurs, then in order for the event Y2 to occur, then they both have to occur. This explains the numerator. Since we know that the event X1 has occurred and we are only interested in a small part of X1 (case where Y2 occurs when X1 has occured), then X1 represents the set of all possibleoutcomes. This explains the denominator.

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