Probability and Problem Solving

if mutually exclusive:
we can put it in the table shown.
we can't find B because of insufficient info

if independent:
\(A = 0.4\)
\(A∪B = A + B - A∩B = 0.8\)
\(A∩B= A*B = 0.4B\)

\(B - 0.4B = 0.4\)
\(0.6B = 0.4\)
\(B = \frac{2}{3}\)

MahmoudFawzy, what basically the equation will be as follows

if mutually exclusive:

P(the target is hit) = P(A)P(B')+P(B)P(A')
as we don't know P(B), we cant determine the answer
am i correct?

Hi, Kindly repost the question according to the rules: https://gmatclub.com/forum/rules-for-po ... 33935.html

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For Independent or Exclusive, the same equation applies:
P(hit) = P(A)(B) + P(A)(B') + P(B)(A')

The difference is in calculating P(A)(B) and P(A')(B'), which can be calculated by multiplication in case of independent but not in Exclusive.

I) What is the probability of Ban hitting the target, if both events are mutually exclusive?
Mutually exclusive—> P(A&B) = 0
—> P(A) + P(B) - P(A&B) = 0.8
—> 0.4 + P(B) - 0 = 0.8
—> P(B) = 0.4

[quote="sark19"]The probability of Alice hitting the target while shooting is 0.4 and probability of either Alice, Ban or both hitting the target is 0.8.
I) What is the probability of Ban hitting the target, if both events are mutually exclusive?
II) What is the probability of Ban hitting the target, if both events are independent?

I wanted to ask if such question should be solved using venn probability or using the expression : P(E)=P(A).P(B') + P(B).P(A') + P(A).P(B). I am not able to derive the answer using the latter. It would be really helpful if someone could answer this query.

Thanks in advance!

It's a strange question setup, and it's not like a GMAT question. The GMAT doesn't ask theoretical questions about "mutually exclusive" events. Instead, the GMAT will propose a real world situation where it's obvious that events are (or are not) mutually exclusive. One issue in this question is that when two people shoot at a target, the outcomes, logically, are not mutually exclusive. So the first question doesn't make a lot of sense. Most of the time, when we talk about mutually exclusive outcomes, we're talking about a single event (not two separate events), and we're discussing two outcomes that cannot both happen. So if you roll a die, the outcome "I roll a 2" and the outcome "I roll an odd number" are mutually exclusive; they can't both happen. Or if you're talking about the weather at a given moment, "It's sunny" and "It's raining" are mutually exclusive, at least if you ignore sun showers. When two events are mutually exclusive, then only three things can happen: the first event, the second event, or neither. The probabilities of those three things must add to 1. This first question would be more natural if it said "the probability it is sunny at a certain moment is 0.4, and the probability it is sunny or raining is 0.8. What is the probability it is raining?" and then the answer is clearly 0.4 (and the remaining 0.2 of the time, maybe it's snowing or some other weather is happening).

The question makes a lot more sense when it discusses 'independent' events; when we talk about independent events, we're normally talking about two (or more) things that happen in succession. Note the difference between independent events and 'mutually exclusive' outcomes: in the mutually exclusive case, we're typically talking only about one event (and we're dividing it into non-overlapping outcomes), but with independent events, we're talking about two or more events. Events are independent when they're not related, probabilistically. That would usually be true when two people shoot at a target; whether the first person hits or misses, that shouldn't affect how likely it is for the second person to hit or miss. When events are independent, you can multiply their probabilities to find the probability they both happen. So here, if p is the probability Ban hits the target, then 0.4*p would be the probability both Alice and Ban hit the target. It's not too convenient to use that to solve this question; instead, it's easier to notice that 1 - 0.8 = 0.2 is the probability they both miss the target. The probability Alice misses is 0.6, and the probability Ban misses is 1 - p, so multiplying, we have

0.6(1 - p) = 0.2
0.6 - 0.6p = 0.2
0.6p = 0.4
p = 2/3

We don't need any Venn diagram principles to solve in either case above. You need Venn principles when events are not independent. That situation can be tested on the GMAT, though it's not tested too often. But if you had a question that said "the probability it is cloudy is 0.4, and the probability it is cold outside is 0.4, what is the probability it is cloudy and cold?" then you're probably dealing with a Venn type problem, because when it's cloudy, it's probably more likely to be cold. The events are potentially related, so are not necessarily independent. You'd need more information to solve this problem: maybe every time it's cloudy it's also cold, and the answer is 0.4, or maybe every time it's cloudy it's not cold for some reason, and the answer could be 0.
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