Three dice are rolled simultaneously. What is the probability that exa

Question

Three dice are rolled simultaneously. What is the probability that exactly two of the dice will come up as the same number?

A. 5/12
B. 11/24
C. 25/54
D. 13/27
E. 1/2

ScottTargetTestPrep · Accepted Answer

There are a total of 6 x 6 x 6 = 216 outcomes.

Let’s say the two numbers that come up are 1 and 2, and then we could have any of the following:

(1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 2, 2), (2, 1, 2) and (2, 2, 1)

However, we have 6C2 = (6 x 5)/2 = 15 ways to choose two distinct numbers from six. For each of these 15 pairs of numbers, we have 6 ways (see the example for 1 and 2 above). Therefore, we have a total of 15 x 6 = 90 ways that exactly two of the dice will come up as the same number. So the probability is 90/216 = 15/36 = 5/12.

Alternate Solution:

Suppose the first die is rolled. The second die can either match the first die (a probability of 1/6) or it can differ from the first die (a probability of 5/6). 
If the second die matches the first die, the only way we can meet the condition of having “exactly two of the dice come up as the same number” is if the last die differs from the first two dice, which has a probability of 5/6. The probability of meeting the condition under this scenario is 1/6 x 5/6 = 5/36.

Alternatively, if the second die differs from the first die, then we can meet the required condition by having the third die match any one of the previous two dice; which is a probability of 2/6 = 1/3. The probability of meeting the condition under this scenario is 5/6 x 1/3 = 5/18.

In total, the probability of meeting the condition is 5/36 + 5/18 = 15/36 = 5/12.

Answer: A

Archit3110 · Answer

for three dice 6^3 = 216
and three dices when rolled , we get 2 same digits
(1,1,2),(1,1,3),(1,1,4)...
this would be 15 times for each digit and 6 digits we would get 15*6 times total
so P
15*6/216
IMO A 
5/12

saur028 · Answer

Probability will be = 1- P(all digits are same)-P( all digits are different)
=1-All digits same ( 6/216)-All digits diff( 3 digit with 6 nos can be arrange in 6*5*4)
=1-6/216-120/216
=90/216=5/12
Ans

BrushMyQuant · Answer

Given that Three dice are rolled simultaneously and We need to find What is the probability that exactly two of the dice will come up as the same number?

As we are rolling three dice =>  Number of cases =  6^3  = 216

Out of the three rolls lets  find out the two rolls in which we will get the same number .
We can get that in 3C2 ways =  \frac{3!}{2!*1!}  = 3 ways

Now these two numbers can be any number out of 6 => First number we can pick in 6 ways and second number we can pick in 1 way as it has to be the same as the first number

For the third number we have 5 ways as it has to be different from the first two

=> Total number of ways = 3 * 6 * 1 * 5

=>  Probability that exactly two of the dice will come up as the same number  =  \frac{3 * 6 * 1 * 5}{6^3}  =  \frac{5}{12}

So,  Answer will be A 
Hope it helps!

Watch the following video to learn How to Solve Dice Rolling Probability Problems

https://www.youtube.com/watch?v=5PVZYQXet18

Three dice are rolled simultaneously. What is the probability that exa

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Three dice are rolled simultaneously. What is the probability that exa

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards