A school employs math teachers and physics teachers. If 4 of these

Question

A school employs math teachers and physics teachers. If 4 of these teachers are selected randomly, what is the probability that at least one math teacher is selected?

(1) The ratio of the number of physics teachers to the number of math teachers is 2 to 1.
(2) The sum of the number of physics teachers and the number of math teachers is 24.

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ManifestDreamMBA · Answer

P(At least one math) = ?

Statement 1
P(Physics) = 2/3
P(Math) = 1/3
P(At least one math) = 1 - P(none math)
= 1 - (2/3)^4
Sufficient

Statement 2
P+M = 24
We don't know the split b/w P and M to know the probability of picking them individually
Insufficient

Answer A

ravi1522 · Answer

Hello ,
Probability for At least one math Teacher can be 1-(probability of no Maths teacher )
As per statement 1
the ratio of both teacher given so we can get the probability as 
1-(2k/(k+2k)^4
Hence sufficient

As per statement 2 total no of teacher is given 
so we cannot find probability as we need count of individual teacher as well

Hence option A is correct

HarshavardhanR · Answer

How do we calculate what is asked?

P (at least 1 math teacher selected) = 1 - P (not even one math teacher selected) = 1 - P (all 4 teachers selected are physics teachers)

Statement 1

P:M = 2:1. So, let P = 2x and M = x. Total number of teachers = 3x.

P (all 4 teachers selected are physics teachers) =  \frac{2x}{3x}  *  \frac{(2x-1)}{(3x-1)}  *  \frac{(2x-2)}{(3x-2)}  *  \frac{(2x-3)}{(3x-3)}

Sidenote:  We can also arrive at the above using  \frac{2xC4 }{ 3xC4}

Observe : we  do not have enough information  to find the above Probability. If we knew x, for instance, we would be able to calculate the exact value of the above. Not the case!

Thus,

P (all 4 selected are physics teachers) -> information not sufficient to calculate this.
So,
P (at least 1 math teacher selected) = 1 - above Probability -> information not sufficient to calculate this.

Statement 1 alone is not sufficient.

Statement 2

Clearly, this statement is also not sufficient. The probability will change based on the exact split of physics and math teachers.

Statements 1 and 2 together

2x + x = 3x = 24 => x = 8.

So,
P -> 16
M -> 8

P (all 4 selected are physics teachers) -> calculable ( \frac{16}{24}  *  \frac{15}{23}  *  \frac{14}{22}  *  \frac{13}{21} )

P (at least 1 math teacher selected) -> calculable (1 -  \frac{16}{24}  *  \frac{15}{23}  *  \frac{14}{22}  *  \frac{13}{21} )

Statements 1 and 2 are together sufficient. Choice C.

---
Harsha

Abhiswarup · Answer

Answer should be C.

No. of Mathematics teacher= x
No. of Physics teacher=2x
 Thus one can't tell how many teachers are for mathematics and how many for physics.
Statement one is not sufficient.

Total teachers is 24, Statement is not sufficient to tell how many are mathematics and how many are physics. Combined we get
Physics Teacher = 16
Mathematics Teacher= 8

Probability of selecting at least one mathematics teacher= (16C3*8C1+16C2*8C2+16C1*8C3+8C4)/24C4

Thus both statements are needed for answering the question.

Correct answer is C

MarkGMATMentor · Answer

This question is a great example of a common pattern in Data Sufficiency:  It looks highly mathematical, but is best solved using reasoning rather than calculations.

Out of 4 selections, we want the probability that at least one math teacher is selected. This is the same as 1 minus the probability that all physics teachers are selected.

Statement 1: 
We know the ratio of physics to math teachers, but not the size of the group. If there were only one teacher selected, you'd know the probability that the teacher is a math teacher is 1/3, and a physics teacher, 2/3.

But with 4 selections, each time a physics teacher is selected, the odds of selecting a math teacher the next time go up and the odds of selecting a physics teacher the next time go down. By how much?

Think about extremes. If the school had 2,000 physics teachers and 1,000 math teachers, the odds of selecting a math teacher wouldn't change much: still about 1/3 every time.

But if the school had only 4 physics teachers and 2 math teachers, the probability of selecting a math teacher goes up significantly each time a physics teacher is selected.

With different possible answers, the statement is  insufficient.

Statement 2:
 With no information as to the relative numbers of physics and math teachers, this statement is clearly  insufficient.

Together:
 With both the total number of teachers and the ratio of physics to math teachers, you'll be able to calculate the actual numbers of each. With complete knowledge of the set, you'll be able to calculate the requested probability. No need to actually do it.  Sufficient.   The answer is C.

The trap here is, of course, choice A.  This trap is best avoided by reasoning; in particular, by thinking about what  would  happen if the number of teachers were very small or very large.

Throughout the GMAT, and particularly on Data Sufficiency, look to reason things through before diving into calculations.

A school employs math teachers and physics teachers. If 4 of these

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GMAT Data Sufficiency (DS) Questions

A school employs math teachers and physics teachers. If 4 of these

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