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.
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