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Problem statement :
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Given two identical unbiased dice, determine the probability of getting sum as 7.

Event = Sum of dots on the top face of both dice is 7.

$E = {(1,6),\ (2,5),\ (3,4),\ (4,3),\ (3,4),\ (5,2),\ (6,1)}$
$|Sample Space|$ = $36$.

Hence, $P(E) = 1/6$

I have a doubt here. As the two dice are given identical, why do we have to consider ordered pairs?
Shouldn't it be unordered consisting of only 3 possible pairs $\{(1,6),\ (2,5),\ (3,4)\} $? Hence, $|S| = 21$ and $P(E) = 3/21$.

taurus05
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    The punchline is that the sample space you describe where order is irrelevant is a valid sample space however it does not satisfy that each outcome in the sample space is equally likely and so you may not simply take the ratio. The correct probability is $\frac{1}{6}$ regardless how well we can distinguish the dice. – JMoravitz May 14 '21 at 13:06
  • @JMoravitz, https://math.stackexchange.com/questions/451579/probability-of-getting-a-certain-sum-of-two-dice-confusion-about-order it says that if we paint a small red dot on one and small blue dot on other die and try out an experiment.... but is it allowed to modify objects in the experiment as per our convenience? – taurus05 May 15 '21 at 05:03
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    yes, of course. Imagine showing your colored dice to a color blind person. They won't know any different. With you both watching the experiment clearly the probability is the same for both of you despite you both having access to different information – JMoravitz May 15 '21 at 05:46
  • @JMoravitz Gotcha! Btw can you recommend me some good online resources to gain much better insights to discrete mathematics? I'm an UG student right now. – taurus05 May 15 '21 at 07:20

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