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# Three dice are rolled simultaneously. What is the

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Three dice are rolled simultaneously. What is the [#permalink]  08 Dec 2017, 20:37
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Question Stats:

75% (12:39) correct 25% (01:16) wrong based on 8 sessions
Three dice are rolled simultaneously. What is the probability that exactly two of the dice will come up as the same number?

A. $$\frac{5}{12}$$
B. $$\frac{11}{24}$$
C. $$\frac{25}{54}$$
D. $$\frac{13}{27}$$
E. $$\frac{1}{2}$$

Drill 2
Question: 12
Page: 526-527

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GRE Prep Club Legend
Joined: 07 Jun 2014
Posts: 4857
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 105

Kudos [?]: 1776 [0], given: 397

Re: Three dice are rolled simultaneously. What is the [#permalink]  29 Dec 2017, 14:12
Expert's post
Explanation

There are a total of 63 = 216 total possible rolls for the three dice.

First figure out the probability of getting exactly two 1’s. There are 5 × 3 = 15 ways this could happen: 112, 113, 114, 115, 116; 121, 131, 141, 151, 161; or 211, 311, 411, 511, 611.

You could repeat this list of 15 possibilities in the obvious way for exactly two 2’s, exactly two 3’s, and so on. Thus, the total number of favorable rolls is 6 × 15 = 90. Because there are 216 possible rolls, 90 of which are favorable, the probability of getting exactly two of the three dice to show the same number is $$\frac{90}{216}=\frac{5}{12}$$, choice A.
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Re: Three dice are rolled simultaneously. What is the [#permalink]  05 Mar 2019, 15:57
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Re: Three dice are rolled simultaneously. What is the [#permalink]  05 Mar 2019, 18:33
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sandy wrote:
Three dice are rolled simultaneously. What is the probability that exactly two of the dice will come up as the same number?

A. $$\frac{5}{12}$$
B. $$\frac{11}{24}$$
C. $$\frac{25}{54}$$
D. $$\frac{13}{27}$$
E. $$\frac{1}{2}$$

Drill 2
Question: 12
Page: 526-527

So,

Total ways = 6*6*6

WITH restrictions
Ways to choose 2 with same number out of 23 = 3C2=3 ways..
These 2 can have any of the 6 sides = 6 ways
The third can take any of the remaining 5, so 5 ways
Total = 3*5*6

Probability = $$\frac{3*5*6}{6*6*6}=\frac{5}{12}$$
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: Three dice are rolled simultaneously. What is the [#permalink]  17 Mar 2019, 18:36
Expert's post
We can also use the complement here. The total number of possibilities is 6*6*6 = 216.

In 6 of these possibilities, all the dice are the same number. That's a probability of $$\frac{6}{216}$$.

What's the probability that none of the dice come up as the same number?

Well, we have a probability of 1 that the first die will be a number. Then we have a $$\frac{5}{6}$$ probability that the next die will not match it. And a $$\frac{4}{6}$$ probability that the third will not match either of the previous two. Together: $$1 \times \frac{5}{6} \times \frac{4}{6} = \frac{20}{36} = \frac{120}{216}$$.

So we have the probability that all the dice are the same, and that none are the same. The only other remaining possibility is that two dice are the same. So:

$$1 - \frac{6}{216} - \frac{120}{216} = \frac{90}{216} = \frac{5}{12}$$
Re: Three dice are rolled simultaneously. What is the   [#permalink] 17 Mar 2019, 18:36
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