There is a 10% chance that Tigers will not win at all during the whole

Question

There is a 10% chance that Tigers will not win at all during the whole season. There is a 20% chance that Federer will not play at all in the whole season. What is the greatest possible probability that the Tigers will win and Federer will play during the season?

(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

OFFICIAL SOLUTION: https://gmatclub.com/forum/there-is-a-1 ... l#p1919727

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Official Answer

E

srikanth9502 · Accepted Answer

The question does not state that these are independent events.
So, the max value of the probabilty will be the lowest of the given two possibilities i.e 80%

This question is discussed in one of the posts of "Probability made easy". Here is the post

P(A or B) = P(A) + P(B) – P(A and B)

P(A) is the probability that event A will occur.

P(B) is the probability that event B will occur.

P(A or B) gives us the union; i.e. the probability that at least one of the two events will occur.

P(A and B) gives us the intersection; i.e. the probability that both events will occur.

Now, how do you find the value of P(A and B)? The value of P(A and B) depends on the relation between event A and event B. Let’s discuss three cases:

1) A and B are independent events

If A and B are independent events such as “the teacher will give math homework,” and “the temperature will exceed 30 degrees celsius,” the probability that both will occur is the product of their individual probabilities.

Say, P(A) = P(the teacher will give math homework) = 0.4

P(B) = P(the temperature will exceed 30 degrees celsius) = 0.3

P(A and B will occur) = 0.4 * 0.3 = 0.12

2) A and B are mutually exclusive events

If A and B are mutually exclusive events, this means they are events that cannot take place at the same time, such as “flipping a coin and getting heads” and “flipping a coin and getting tails.” You cannot get both heads and tails at the same time when you flip a coin. Similarly, “It will rain today” and “It will not rain today” are mutually exclusive events – only one of the two will happen.

In these cases, P(A and B will occur) = 0

3) A and B are related in some other way

Events A and B could be related but not in either of the two ways discussed above – “The stock market will rise by 100 points” and “Stock S will rise by 10 points” could be two related events, but are not independent or mutually exclusive. Here, the probability that both occur would need to be given to you. What we can find here is the range in which this probability must lie.

Maximum value of P(A and B):

The maximum value of P(A and B) is the lower of the two probabilities, P(A) and P(B).

Say P(A) = 0.4 and P(B) = 0.7

The maximum probability of intersection can be 0.4 because P(A) = 0.4. If probability of one event is 0.4, probability of both occurring can certainly not be more than 0.4.

Minimum value of P(A and B):

To find the minimum value of P(A and B), consider that any probability cannot exceed 1, so the maximum P(A or B) is 1.

Remember, P(A or B) = P(A) + P(B) – P(A and B)

1 = 0.4 + 0.7 – P(A and B)

P(A and B) = 0.1 (at least)

Therefore, the actual value of P(A and B) will lie somewhere between 0.1 and 0.4 (both inclusive).

Now let’s take a look at a GMAT question using these fundamentals:

There is a 10% chance that Tigers will not win at all during the whole season. There is a 20% chance that Federer will not play at all in the whole season. What is the greatest possible probability that the Tigers will win and Federer will play during the season?
(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%

Let’s review what we are given.

P(Tigers will not win at all) = 0.1

P(Tigers will win) = 1 – 0.1 = 0.9

P(Federer will not play at all) = 0.2

P(Federer will play) = 1 – 0.2 = 0.8

Do we know the relation between the two events “Tigers will win” (A) and “Federer will play” (B)? No. They are not mutually exclusive and we do not know whether they are independent.

If they are independent, then the P(A and B) = 0.9 * 0.8 = 0.72

If the relation between the two events is unknown, then the maximum value of P(A and B) will be 0.8 because P(B), the lesser of the two given probabilities, is 0.8.

Since 0.8, or 80%, is the greater value, the greatest possibility that the Tigers will win and Federer will play during the season is 80%. Therefore, our answer is E

Skywalker18 · Answer

There is a 10% chance that Tigers will not win at all during the whole season 
We can infer that there is 90% chance Tigers will win .
Similarly There is a 20% chance that Federer will not play at all in the whole season
We can also infer that there is 80% chance that Federer will play.

Answer E

davedekoos · Answer

The question is asking what the chance is that BOTH events occur (Tigers win AND Federer plays). To find this we need the respective probabilities of each event.
Chance that the Tigers do not win = 10%. This means that the chance that the Tigers win at least once is 1-10% = 90%
Chance that Federer doesn't play = 20%. This means that the chance that Federer does play is 1-20% = 80%

So the probability of BOTH events occurring is 90% * 80% = 72%

Answer D

Unless I'm misinterpreting the question, since I find the wording a little strange. It is redundant and ambiguous to say the "maximum possible probability" in a situation where the probability is not a dependent variable. Here we are told exactly what the probabilities are, so in my interpretation of the question, it should have simply asked "What is the probability that the Tigers will win and Federer will play during the season?"

bethebest · Answer

I do not understand this completely,i'm afraid.!! 
I attempted this problem in the following way

Tigers will not win = 10%, which means Tigers will win =90% = 9/10
Federer will not play = 20% which means Federer will play = 80% = 80/100 = 4/5

Probability of both occurring is 9/10*4/5 =18*25 = 72%

But this is the wrong answer. Are we assuming that the GREATEST probability of both occurring is Federer wins all the games that he plays (80%)? Please suggest.

KbSharma · Answer

I was about to go with simple .9 * .8 approach but then the keyword not at all and the question will comes into picture.
So IMO, it should be 1-((prob of not playing)*(prob of not wining)),
which is 1-((.2)*(.1)) = 80%

chetan2u · Answer

Hi,
 you are correct that the probability for two INDEPENDENT events are P(A)*P(B) 
and here it should be 72%..

May be the Q means that the greatest possiblity where you have control over happening of events..

So here, the entire playing by Federer happens when Tigers wins..
In other terms Federer playing is a subset of Tigers wins..
so greatest possiblity will be equal to the lesser of two, which is 80%

jokschmer · Answer

Bunuel,

can you please tell exact logic behind this question and how we detect this type of question to apply correct logic.

FB2017 · Answer

Got it wrong on the first go. Had considered each event to be independent and left out the greatest possibility aside and came up with
(prob of roger playing)*(prob of tiger wining)
.9*.8=.72*100=72%

However below is correct
1-((prob of federer not playing)*(prob of tiger not wining)),
which is 1-((.2)*(.1)) = .8*100=80%
 Bunuel  can you please confirm if "greatest possibility" in Problem statement is the keyword to identify the right solution here.
Please advise.

SeanWtx · Answer

I would really hope that the GMAT would include more information on how/whether the two events are related if they put this on an actual exam. We don't even know if the Tigers and Federer play the same sport (and note how the question uses a well-known baseball team name with a well-known tennis player's name to further lead one to believe they are playing in different leagues). I get that it says "greatest probability," but I think at least mentioning that Federer plays for the Tigers is a basic piece of information that would make the question a fair one to ask.

sahilvijay · Answer

Hi  FB2017

i can see a mistake in your calculation ... which is 1-0.2 = 0.8 , whereas 1-(0.2x0.1) is 1-0.02 = 0.98 ?????

----------------------------------------------
Hello  Bunuel

what is wrong in the concept which i applied
P(win) =0.9
P(play) = 0.8
2 independent events , P ( win and play) = 0.9x0.8 = 0.72 
is this probability of both events ( independent) occuring is wrong??? kindly advise. or the OA is wrong?

Bunuel · Answer

Veritas Prep Official Solution:

Let’s review what we are given.

P(Tigers will not win at all) = 0.1

P(Tigers will win) = 1 – 0.1 = 0.9

P(Federer will not play at all) = 0.2

P(Federer will play) = 1 – 0.2 = 0.8

Do we know the relation between the two events “Tigers will win” (A) and “Federer will play” (B)? No. They are not mutually exclusive and we do not know whether they are independent.

If they are independent, then the P(A and B) = 0.9 * 0.8 = 0.72

If the relation between the two events is unknown, then the maximum value of P(A and B) will be 0.8 because P(B), the lesser of the two given probabilities, is 0.8.

Since 0.8, or 80%, is the greater value, the greatest possibility that the Tigers will win and Federer will play during the season is 80%. Therefore, our  answer is E .

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