JEE Advanced 2017 Paper 2 · Mathematics · Probability · medium

Three fair six-faced dice are rolled together.

One or more options correct

Three fair six-faced dice are rolled together. Which of the following is/are correct?

A. The probability that the largest number shown is exactly $4$ is $\dfrac{37}{216}$
B. Given that the sum of the three numbers is $10$ , the probability that the largest number is $4$ is $\dfrac29$
C. Given that the sum is $9$ , the probability that all three numbers are different is $\dfrac35$
D. Given that every number is at least $2$ , the probability that the sum is $9$ is $\dfrac19$

Show answer and solution

Answer: A, B

Enumerate the

216

ordered outcomes. (A)

P(\max\le4)-P(\max\le3)=\frac{64-27}{216}=\frac{37}{216}

; true. (B) Sum

10

has

27

ordered triples; those with maximum

4

are the permutations of

(4,4,2)

and

(4,3,3)

, i.e.

3+3=6

, giving

\frac{6}{27}=\frac29

; true. (C) Sum

9

has

25

ordered triples; the all-different ones come from

\{1,2,6\},\{1,3,5\},\{2,3,4\}

, giving

18

ordered triples, so the probability is

\frac{18}{25}\ne\frac35

; false. (D) With all numbers in

\{2,\dots,6\}

there are

125

outcomes and the sum-

9

ones are

(2,2,5)

and

(2,3,4),(3,3,3)

types giving

10

outcomes, i.e.

\frac{2}{25}\ne\frac19

; false.

Original question written in the JEE Advanced 2017 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake