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If a die is rolled 10 times. What is the probability that the sum of the results is less than or equal to 20?

I was trying to solve this using something like $P(X_1 + X_2 + ....+X_{10} \le 20)$ but was not able to proceed. And, if I try to work out individual cases, it will take a lot of time.

meta_finance
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    Use the multinomial distribution – Peter Jul 09 '16 at 20:52
  • @Peter I am not sure how to use multinomial distribution as I have just used multinomials for partitions and not for sum of random variables. – meta_finance Jul 09 '16 at 21:01
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    If we want an approximation, and are not too fussy, we can use the normal. This should not be too bad, particularly if we use a continuity correction, and find the probability that the appropriate normal is $\le 20.5$. – André Nicolas Jul 09 '16 at 21:08
  • For each partition, the multinomial distribution gives the probability that this particular partition occurs. – Peter Jul 09 '16 at 21:08
  • @AndréNicolas I do not think that the normal approximation is very well here. But let's see ... – Peter Jul 09 '16 at 21:10
  • @Peter: Nowadays one jusr writes a quick program. But in the old days, for a ballpark estimate, one would stretch the normal approximation beyond the zone of comfort. – André Nicolas Jul 09 '16 at 21:14
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    Is it an homework ? May be you are asked to give an approximate answer only : in this case, use a normal approximation N(m=35,\sigma^2=17.5) which gives you something at more than $3 \sigma$ at the left from the mean, otherwise said, less than $0.5$ percent chance of occurrence (less than 1 out of 200 cases are favorable) – Jean Marie Jul 09 '16 at 21:16
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    I get the exact result : $0.00289$. What does the normal approximation give ? – Peter Jul 09 '16 at 21:23
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    @Peter I used the normal distribution as Andre and Jean suggested and the answer I came up with is $0.0030$. So, I guess it is pretty close. – meta_finance Jul 09 '16 at 21:26
  • Did not expect this. Really close. – Peter Jul 09 '16 at 21:30
  • @Peter I am guessing the difference got lost when I converted the fraction to decimal and using the normal table upto only two decimal points. – meta_finance Jul 09 '16 at 21:32

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