Let A₁, A₂, A₃ be events satisfying 0 < P(A_i) < 1 for i = 1, 2, 3. Which of the following statements is true? 

Concept:

Conditional Probability:

For two events A and B, the conditional probability of A occurring given that B has occurred is denoted as

, provided that P(B) > 0.

P(A_3 | A_2) $P(A_3 | A_2)$ Explanation:

Option 1:

Counter example: 

Let, 

  ,   

 ,   and  

 ,  and
 

Left-hand side:   

Right-hand side:  0.1667 

Here, the left-hand side 0.4167 is greater than the right-hand side 0.1667, which violates the inequality. 

Thus, this provides a counter example where

So, option 1) is false.

Option 2:

Let us define three events  in a probability space with the following probabilities:

 , 

 , 

Left-hand side: 

Right-hand side: 

Hence, 

So option 2) is false.

Option 3:
   
This statement looks plausible because the probability of the union of two events is generally less than

or equal to the sum of the individual probabilities. This would make the product of the conditional probabilities

potentially greater than or equal to the union probability. Hence, this option is true.

Option 4:
 

Left-hand side: 

Right-hand side: 

The correct answer is Option 3).