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Thread: Symmetric 6 sided dice, exactly 5 sixes; chance of six > 0.95

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    Symmetric 6 sided dice, exactly 5 sixes; chance of six > 0.95

    Hello, I'm having problems with probability and I hope You can help me! (If this is more suitable for uni level, please move it there)

    My problem is : A symmetric dice is rolled 30 times. What is the probability that a 6 turns up exactly 5 times?

    My logic:
    We define
    $\displaystyle \Omega = \{ (a_1, a_2, ... , a_{30} ) | a_i \in {1,...,6}, i=1,..,30 \}$
    Obviously cardinality of $\displaystyle \Omega$ is $\displaystyle k(\Omega)=6^{30}$.
    Now we focus of cardinality of $\displaystyle A = \{"exactly \,5 \, sixes" \}$. In order to have exactly 5 sixes, we need to have 5 sixes in $\displaystyle (a_1,...,a_{30})$ and rest can be from 1 to 5. So we have $\displaystyle 5^{25}$ choices for rest. When we selected numbers, we can "move" block of sixes in any way we want (since it doesn't matter if 5 sixes felt in first 5 throws or last), we can do that in 25 ways*. So $\displaystyle k(A)=25 \cdot 5^{25}=5^{27}$ and we get probability by dividing those two: $\displaystyle P(A)=\frac{5^{27}}{6^{30}}$

    *The number of r-permutations of set with n elements is $\displaystyle \frac{n!}{(n-r)!}$, we can consider here a block of 5 sixes as one big block. By this logic we can choose block positions in $\displaystyle \frac{25!}{24!}$, which would be my proof for above selection.

    Second problem: How many dice you need to throw in order to have more then 0.95 chance of six.

    I do not have a clear idea how to solve this one, my first thought was calculating chance of complement (i.e. chances of numbers 1 to 5 < 0.5), so basically I'm looking at $\displaystyle (\frac{5}{6})^n = 0.05$ and I get n has to be over 16. My problem is how to solve this in more strict way.

    Thank You for Your time.
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    Re: Symmetric 6 sided dice, exactly 5 sixes; chance of six > 0.95

    Quote Originally Posted by magicka View Post
    A symmetric dice is rolled 30 times. What is the probability that a 6 turns up exactly 5 times?
    This is binomial: $\displaystyle \binom{N}{k}\left( p\right)^k\left(1-p\right)^{N-k}$.

    Now for this problem: $\displaystyle \binom{30}{5}\left(\frac{1}{6}\right)^5\left(\frac {5}{6}\right)^{25}$
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