In the town of Z, the town lion roars on some days and not on others.

You are looking at it incorrectly. Nowhere do the statements say anything about a situation when the lion did not roar and it did not rain. You can not assume that x + x-10 = 31 . It might be that x=13, x-10=3 and thus on 31- (13+3) = 15 days nothing happend (neither the lion roared nor it rained!). Even this set of 13,3 satisfies both the statements and the question stem.

You are correct in assuming that 20, 10 can be the number of days but will it always be like this? No.

Hope this answers your question.

We have a total of 30 days in march. To calculate the probability of "either roaring or raining" we need to have the number of days the lion roared and the number of days of rain.

Statement 1: Insufficient (see above). No information about number of days.
Statement 2: Insufficient (see above). No information about number of days it rained.

Statement 1+2: Still not sufficient. I am looking to fill this formula: days roaring / 30 + days raining / 30.

Answer E

We require the following information to answer this question:
1) On how many days in March, Lion roared
2) On how many days it rained in March
3) On how many days Lion roared as well as it rained

Question : Probability Either Lion roared or it rained = ?

Statement 1: last March, the lion never roared on a rainy day.

This provides us information No. 3 but Information 1 and @ are still unknown, Hence
NOT SUFFICIENT

Statement 2: Last March, the lion roared on 10 fewer days than it rained.

There are 31 days in March
No. of Days Lion Roared = x
i.e. No. of Days it rained = x+10
But Information 3 is still unknown and the value of x also is unknown hence
NOT SUFFICIENT

Combining the two statements
No. of Days Lion Roared = x
i.e. No. of Days it rained = x+10
No day lion roared and It rained
But x+x+10 can be anything less than or equal 31, therefore,
NOT SUFFICIENT

Answer: Option E

Target question: What is the probability that on that day, either the town lion roared or it rained?
This is a good candidate for rephrasing the target question.

This is an OR probability. The OR probability rule says, P(A or B) = P(A) + P(B) - P(A and B)
So, P(rained or roared) = P(rained) + P(roared) - P(rained and roared). So . . .

REPHRASED target question: What is the value of P(rained) + P(roared) - P(rained and roared)?

Aside: Here’s a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

Statement 1: Last March, the lion never roared on a rainy day.
In other words, P(rained and roared) = 0
Since we still don't know the values of P(rained) and P(roared), we cannot evaluate P(rained) + P(roared) - P(rained and roared)
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Last March, the lion roared on 10 fewer days than it rained.
Let x = # of days the lion roared
So, x+10 = # of days it rained
This means P(roared) = x/31 and P(rained) = (x+10)/31
Since we still don't know the actual values of P(rained) and P(roared), and we don't know the value of P(roared and rained) we cannot evaluate P(rained) + P(roared) - P(rained and roared)
Since cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
From statement 1 we know that P(rained and roared) = 0
From statement 2 we know that P(roared) = x/31 and P(rained) = (x+10)/31
Put them together and we get: P(rained or roared) = (x+10)/31 + x/31 - 0
Since we still cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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