A certain experiment has three possible outcomes. The outcomes are mut

Question

A certain experiment has three possible outcomes. The outcomes are mutually exclusive and have probabilities x, y, and z, respectively. What is the value of x?

(1) The ratio x:y:z = 4:2:1
(2) x^2 + xy + xz - x - y - z = -3/7

M36-108

Bunuel · Accepted Answer

Official Solution:

A certain experiment has three possible outcomes. The outcomes are mutually exclusive and have probabilities of  x ,  y , and  z , respectively. What is the value of  x ? 
 
Since the experiment has three mutually exclusive outcomes, then  x+y+z=1 .
 
(1) The ratio  x:y:z = 4:2:1 
 
We can write  4k+2k+k=1 . This gives  k=\frac{1}{7}  and thus  x=4k=\frac{4}{7} . Sufficient.
 
(2)  x^2 + xy + xz - x - y - z = -\frac{3}{7} :
 
 x(x + y + z) - (x + y + z) = -\frac{3}{7} 
 
Substitute  x+y+z=1  into the above:  x - 1 = -\frac{3}{7} . This gives  x=\frac{4}{7} . Sufficient.

Answer: D

acegmat123 · Answer

We know that sum of probabilities of all outcomes is "1"
i.e. x+y+z=1

1. x = 4/7

2. (x-1)(x+y+z) = -3/7

x-1 = -3/7

x = 4/7

So answer is D

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CrackverbalGMAT · Answer

This is a question which tests you on your knowledge of probability. But, it also tests you on your ability to interpret ratios and simplify algebraic expressions.

Mathematically, probability is defined as,

Probability (Event) =  \frac{No. of favourable outcomes}{Total possible outcomes} 
.
In this problem, since there are only three possible outcomes and these are also mutually exclusive, the sum of the probabilities has to be 1, i.e.
 x + y + z = 1
This is the data that we can glean from the question statements.

Now, from statement I alone, since x, y and z are in the ratio of 4:2:1, let us assume their actual values to be 4k, 2k and k respectively. The total of these three terms comes out to be 7k. 
Therefore, x =  \frac{4k}{7k} 
Or , x =  \frac{4}{7} .
So, statement I alone is sufficient to find the value of x. The possible answers are A or D. Options, B, C and E can be eliminated.

From statement II alone, we have this complex looking expression (or is it?):
 x^2  + xy + xz - x - y - z = - \frac{3}{7} .

Let us try and simplify the LHS of the above equation as much as we can.
 x^2 + x (y + z) – (x+y+z) = - \frac{3}{7}

From the question data, we know x + y + z = 1. Substituting in the above equation, we have,

x^2  + x(y+z) – 1 = - \frac{3}{7}

Taking x common on the LHS after transposing the 1 onto the RHS, we have,

x ( x + y + z) = 1 –  \frac{3}{7}

Substituting the value of (x + y + z) and simplifying, we have, 
x =  \frac{4}{7} 
Statement II alone is also sufficient to find the value of x. Answer option A can be ruled out. The correct answer option is D.

Interpreting the first statement is easy, but, for some, the expression on the LHS in Statement II can look overwhelming. The best thing to do is to take it one step at a time and simplify the expression as much as possible before simplifying.

Obviously, knowing the basic concepts of Probability helps.

Hope this helps!
Thanks.

svsivakrishna · Answer

Given x, y and z are probabilities of mutually exclusive events, therefore x+y+z=1;

Question: to find x.

and

Statement (1) is: The ratio x:y:z = 4:2:1

Since we know x+y+z=1 and the ratio of x:y:z = 4:2:1, we can find the value of x. Theferore statement I is sufficient.

Statement (2) is:x^2 + xy + xz - x - y - z = -3/7

Simplifying the above equation,
x(x+y+z)-1(x+y+z)=-3/7

(x-1)(x+y+z)=-3/7

But we know that x+y+z=1

So, x-1=-3/7, x=4/7. Theferore statement II is sufficient.

Option D

ManishaPrepMinds · Answer

Since the outcomes are mutually exclusive and exhaustive
x + y + z = 1

Statement (1) Analysis: 
The ratio x:y:z = 4:2:1
x = 4/7 (1) = 4/7
 Statement (1) alone: SUFFICIENT

Statement (2) Analysis: 
x^2 + xy + xz - x - y - z = -3/7
x(x +y +z) - (x +y +z) =-3/7
(x-1) (x +y +z) =-3/7
(x-1) =-3/7
x = 4/7
 Statement (2) alone: SUFFICIENT

Answer: D. EACH statement ALONE is sufficient

These are the fundamental mistakes students make. 
 Error:  Starting with individual statements without establishing x + y + z = 1
 Must Do First:  "Mutually exclusive and exhaustive" → x + y + z = 1

Error:  Trying to solve Statement (2) with algebra without recognizing the factoring opportunity.
 Must Do First:  Always look for common factors before expanding!

BrenCraw · Answer

This is so helpful! Thank you!!

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