GMAT Math

There is an example in OG Edition 13 Page 120 for Discrete Probability that goes as...

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

Step 1 and Step 2 are obviously clear...

Can someone explain step 3? It doesn't make any sense to me at all...

"it can be determined that A and C are "not mutually exclusive" since P(A)+P(C)=1.08" - All this means is that P(A) + P(C) > 1 but how can you infer that A and C ARE NOT MUTUALLY EXCLUSIVE based on this fact.(Is this a property/rule for 2 events that are not mutually exclusive?)

Furthermore it says
"which is greater than 1, and therefore cannot equal P(A or C)" - What???How???Why???

then

"from this it follows that P(A and C) >= 0.08" - How?(I tried to put it this way but did not arrive at any meaningful conclusion. - P(A and C)>=1.08-1 which is P(A)+P(C) - 1.

then

"Once can also deduce that P(A and C) <= P(A) = 0.23 , since A∩C is a subset of A" - How?(and why is it not <=P(C) and how is A∩C is a subset of A?)

then

and that P(A or C) >= P(C) = 0.85 since C is a subset of A∪C - How???Why????

Finally,
"Thus, one can conclude that 0.85 <= P(A or C) <= 1 and 0.08 <= P(A and C) <= 0.23 " - How???

There is something NOT right with this example or some subtle fact/property that I am missing to understand.
Any help is greatly appreciated.

Thanks in advance

Thanks for replying...
you did explain the concept of P(anything)>1 cannot exist.
Based on this fact,

1)If A and C were mutually exclusive then P(A and C) would be 0 and P(A or C) = P(A)+P(C) - which comes to 1.08 in this case which is simply not possible and A and C are not mutually exclusive and therefore P(A and C) > 0
2)The equation - P(A or C) = P(A)+P(C) - P(A and C) holds true for A and C
therefore P(A or C) Not Equal P(A)+P(C) (as per the above equation - P(A and C) has some value)
3)Since P(A or C) = 1.08[P(A)+P(C)] - P(A and C) therefore P(A and C) has to be greater than or equal to .08 since only such a value can satisfy P(A or C) < 1 - Based on the fact that P(Anything) is always < 1.
4)P(A and C) <= P(A) = 0.23 because Between A and C,P(A) is least and Probability of 2 events happening together will be lesser than the probability of the event with least probability among the two(this can be proved with the example you gave)
5)Similarly - P(A or C) >= P(C) = 0.85 because Between A and C,P(C) is higher and Probability of either 1 event happening from 2 events is higher(or equal to) the probability of more probable(higher probability) event.

Did I understand this correctly?
Thanks so much again for replying back.

Hi gmat 72014,

This explanation of OG (if it is the way you have written here - i do not have OG currently) assumes that reader already knows a few basic things.

I will restate the whole problem and then try to tell you. Hope you will understand.

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

Now firstly u should know that p(A or C) = p(A) + p(C) - p(A or C)
We do not have the values of both p(A or C) and p(A and C) but we can find out the range in which these values may lie.
So we start adding p(A) and p(C) : 0.23 + 0.85 = 1.08 . But we know that the probability of any event can never exceed 1 so 'p(A) + p(C) - p(A and C)' cannot equate to p(A or C).
One of your question about how we concluded that A and C are not mutually exclusive: this is because the sum of probabilities of mutually exclusive events is always equal to 1. In this case p(A) + p(C) = 1.08 that is more than 1. So A and C are not mutually exclusive.

Till above point we did not take p(A and C) into consideration, now if we consider its value then we can say that in order to find out the probability of occurrence of either A or C we need to have p(A and C) as greater than 0.08. This is because p(A) + p(C) = 1.08 and since the value of p(A or C) cannot be more than 1 so we need to subtract some value of p(A and C) that is either equal to or greater than 0.08.

Next we come to the point that p(A and C) is a subset of p(A).
This can be very well understood with the help of Venn Diagrams, these are nothing but overlapping circles. Consider two circles, one circle as A and one circle as C, and these two circles overlap each other. So the portion of both the circles that overlaps is called the event 'A and C' this portion is common to both A and C so this will be the p(A and C). Now since this portion is common to both A and C thus it will be included in both A and C and so p(A and C) is a subset of p(A). And hence it should necessarily be smaller than p(A). P(A and C) should be less than or equal to 0.23.
When you consider (A or C), this set covers all the area of set A and set C. So it is clear that C will definitely be a smaller portion as compared to the portion of A and C combined (that is the set A or C). So C will be a subset of (A or C). And thus p(A or C) will be greater or equal to p(C).
So we can conclude the range of p(A or C) as between 0.85 and 1 and the range of p(A and C) as between 0.08 and 0.23.

Moreover for your better understanding of the circles A and C i am just attaching a pic of a self made diagram.