Consider an experiment with events A, B, and C for which P(A) = 0.23, P(B) = 0.40, and P(C) = 0.85. Also, suppose that events A and B are mutually exclusive and events B and C are independent. Then
Step 1: P(A or B) = P(A)+P(B) (Since A and B are mutually exclusive)
= 0.23+0.40
= 0.63
Step 2: P(B or C) = P(B)+P(C)-P(B)P(C) (by independence)
= 0.40+0.85-(0.40)(0.85)
= 0.91
Step 3: P(A or C) and P(A and C) cannot be determined using the information given. But it can be determined that A and C are "not mutually exclusive" since P(A)+P(C)=1.08, which is greater than 1, and therefore cannot equal P(A or C); from this it follows that P(A and C) >= 0.08. Once can also deduce that P(A and C) <= P(A) = 0.23, since A∩C is a subset of A, and that P(A or C) >= P(C) = 0.85 since C is a subset of A∪C. Thus, one can conclude that 0.85 <= P(A or C) <= 1 and 0.08 <= P(A and C) <= 0.23
I do understand Step 1 and Step 2 but Step 3 I have difficulties understanding it. I also understand the part that A and C are "not mutually exclusive" but the rest of the statement beginning from P(A and C)>=0.08, I fail to comprehend. I tried hard but I cannot really understand that explanation.
Can someone please throw some light on this? I am just beginning to familiarize myself with Discrete Probability (which I am extremely bad at).
Source: GMAT Quant Review 2nd Edition
Step 1: P(A or B) = P(A)+P(B) (Since A and B are mutually exclusive)
= 0.23+0.40
= 0.63
Step 2: P(B or C) = P(B)+P(C)-P(B)P(C) (by independence)
= 0.40+0.85-(0.40)(0.85)
= 0.91
Step 3: P(A or C) and P(A and C) cannot be determined using the information given. But it can be determined that A and C are "not mutually exclusive" since P(A)+P(C)=1.08, which is greater than 1, and therefore cannot equal P(A or C); from this it follows that P(A and C) >= 0.08. Once can also deduce that P(A and C) <= P(A) = 0.23, since A∩C is a subset of A, and that P(A or C) >= P(C) = 0.85 since C is a subset of A∪C. Thus, one can conclude that 0.85 <= P(A or C) <= 1 and 0.08 <= P(A and C) <= 0.23
I do understand Step 1 and Step 2 but Step 3 I have difficulties understanding it. I also understand the part that A and C are "not mutually exclusive" but the rest of the statement beginning from P(A and C)>=0.08, I fail to comprehend. I tried hard but I cannot really understand that explanation.
Can someone please throw some light on this? I am just beginning to familiarize myself with Discrete Probability (which I am extremely bad at).
Source: GMAT Quant Review 2nd Edition
