JEE Main · 2016 · 03 Apr 2016 · Offline · Probability

Let two fair six-faced dice A and B be thrown simultaneously. If

E_{1}

is the event that die A shows up four,

E_{2}

is the event that die B shows up two and

E_{3}

is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

Answer: (C)

Solution

The single-event probabilities are

P (E_{1}) = \frac{1}{6}

P (E_{2}) = \frac{1}{6}

and

P (E_{3}) = \frac{1}{2}

. Each pair is independent:

P (E_{1} \cap E_{2}) = \frac{1}{36} = P (E_{1}) P (E_{2})

, and likewise

P (E_{1} \cap E_{3}) = P (E_{2} \cap E_{3}) = \frac{1}{12}

equal their products. But all three cannot happen together, since A

= 4

and B

= 2

give sum 6, which is even, so

P (E_{1} \cap E_{2} \cap E_{3}) = 0

while

P (E_{1}) P (E_{2}) P (E_{3}) = \frac{1}{72} \neq = 0

. The events are pairwise but not mutually independent, so the claim that

E_{1}

E_{2}

and

E_{3}

are independent is the untrue one.

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option