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I tried solving this question:

Toss a die 60 times, what is the probability that the number of sixes is a) greater or equal 12, b) strictly between 8 and 12

My solution for part a was P(grt or equal 12) = 1 - P(less than 12) = 1 - (11 * 1/6)/60 but i don't think it's correct. I don't have any idea about part b.

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(a) $$\sum_{k=12}^{60}\binom{60}{k}\left(\frac{1}{6}\right)^{k}\left(\frac{5}{6}\right)^{60-k}=1-\sum_{k=0}^{11}\binom{60}{k}\left(\frac{1}{6}\right)^{k}\left(\frac{5}{6}\right)^{60-k}.$$

(b) $$\sum_{k=9}^{11}\binom{60}{k}\left(\frac{1}{6}\right)^{k}\left(\frac{5}{6}\right)^{60-k}.$$

Explanation: suppose that you are looking for the probability of the number of sixes being exactly $k$ ($k$ integer, $0\leq k\leq60$). Then, the event of a six, which has probability $1/6$, must occur exactly $k$ times, whereas the complementary event, whose probability is $5/6$, must occur $60-k$ times. Finally, the number of ways in which you can choose which tosses out of the 60 tosses are to result in $6$ is $\binom{60}{k}$.

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  • $\begingroup$ Thanks, can you explain a little about this formula or link me to it? $\endgroup$ – Nima G Sep 17 '13 at 18:15
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    $\begingroup$ This formula explains that there are (60 choose k) ways to pick the 12 sixes that occur because there are 60! ways to enumerate the possiblities of 60 die rolls, and you have to divide out the equivalent orderings of 12 sixes. The rest is explained in this link: mathwords.com/b/binomial_probability_formula.htm $\endgroup$ – user85362 Sep 17 '13 at 18:18
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The answer for $(a)$ is approximately 0.292056, by the work performed in the above answer. The answer for $(b)$ is approximately 0.395897 by the work performed in the above answer.

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