The numbers a, b, and c are chosen at random, with replacement, from

Question

The numbers a, b, and C are chosen at random, with replacement, from the set of integers from 1 to 20, inclusive. What is the probability that ab − c is odd?

A. 1/8
B. 1/4
C. 3/8
D. 1/2
E. 3/4

pushpitkc · Accepted Answer

These are the 8 possibility for the variables A,B, and C

--A---B---C----AB - C ---
--O---O---O-----Even----
--O---O---E-----Odd-----
--O---E---O-----Even----
--O---E---E-----Odd-----
--E---O---E-----Even----
--E---O---O-----Odd-----
--E---E---E-----Even----
--E---E---O-----Odd-----

Out of the 8 possibilities, 4 possibilities have AB-C as Odd.

Therefore, the probability that ab-c is odd is  \frac{4}{8}  or  \frac{1}{2}  (Option D)

baru · Answer

These are the 4 possibility for the variables AB, and C

AB - C ---
 O & O = E
 E & E = E
 O & E = O
 E & O = O

So , P(Odd) = 2/4 = 1/2

GMATWhizTeam · Answer

Solution 
 • From 1 to 20 there are 20 integers in which 10 are odd and 10 are even.
 o The probability of picking an even number P(E) = The probability of picking an odd number P(O)  = \frac{1}{2}  
• The following table shows the cases when ab – c will be odd and the probability of the corresponding cases:
 https://www.dropbox.com/s/kzuecnvwhmlqujj/26_07_table.jpg?dl=1 
• Thus, P(ab – c is odd)  = 4*\frac{1}{2}*\frac{1}{2} *\frac{1}{2} = \frac{4}{8} = \frac{1}{2}   
Thus, the correct answer is  Option D.

Kinshook · Answer

Given:The numbers a, b, and C are chosen at random, with replacement, from the set of integers from 1 to 20, inclusive. 
Asked: What is the probability that ab − c is odd?

a  b  c  ab-c  
  EVEN  EVEN  EVEN  EVEN  
  EVEN  EVEN  ODD  ODD  
  EVEN  ODD  EVEN  EVEN  
  EVEN  ODD  ODD  ODD  
  ODD  EVEN  EVEN  EVEN  
  ODD  EVEN  ODD  ODD  
  ODD  ODD  EVEN  ODD  
  ODD  ODD  ODD  EVEN

Probability that ab-c is odd =  \frac{4}{8} = \frac{1}{2}

IMO D

EgmatQuantExpert · Answer

Solution 
 Given

• There is a set with integers from 1 to 20.
• A, B, C are three numbers picked up randomly with replacement.

To Find 
 • the probability that AB – C is odd.

Approach and Working Out

We need not find the number of cases here.

A logical way is to understand that we are not biased towards odd or even (as we have 10 each) and thus AB – C is equally likely to become an odd number or an even number.

Let’s see what I mean by that.

• At any moment there are 10 odd numbers and 10 even numbers as we are replacing the picked-up numbers. 
  o AB – C can be odd if,
  AB is even and C is odd or
 AB is odd and C is even.   
 • Once we have picked up A and B,
 o We already know the odd-even nature of AB.
o If AB is odd, then we should pick an even number for C.
  The probability is ½.  
 o If AB is even, then we should pick an odd number for C.
  Again, the probability is ½.

Correct Answer: Option D

Adarsh_24 · Answer

This could be an alternative approach.

There is an equal number of even and odd possible number and everyone has equal chance.

Given ab-c = odd
Then ab and c opposite parity
1)
ab even => 3/4 chance (ee, oe, eo)
c odd => 1/2
2)
Ab odd => 1/4(oo)
C even => 1/2

3/8 + 1/8 = 1/2

MartyMurray · Answer

The numbers a, b, and C are chosen at random, with replacement, from the set of integers from 1 to 20, inclusive. What is the probability that ab − c is odd?

There are 4 ways in which ab - c can be odd.

odd × odd - even

even × even - odd

even × odd - odd

odd × even - odd

Since the numbers a, b, and c are chosen at random with replacement from a set of integers, 1 to 20, such that half are even and half are odd, for each of those numbers chosen, the probability that that number will be even is 1/2, and the probability that that number will be odd is 1/2.

So, the probability of each of the above 4 favorable outcomes is 1/2 × 1/2 × 1/2 = 1/8.

4 × 1/8 = 1/2

A. 1/8
B. 1/4
C. 3/8
D. 1/2
E. 3/4

Correct answer:   D

The numbers a, b, and c are chosen at random, with replacement, from

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a	b	c	ab-c
EVEN	EVEN	EVEN	EVEN
EVEN	EVEN	ODD	ODD
EVEN	ODD	EVEN	EVEN
EVEN	ODD	ODD	ODD
ODD	EVEN	EVEN	EVEN
ODD	EVEN	ODD	ODD
ODD	ODD	EVEN	ODD
ODD	ODD	ODD	EVEN

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The numbers a, b, and c are chosen at random, with replacement, from

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