The probability of rolling 4 dice and getting a 6.

Question

The probability of rolling 2 dice and getting a 6 on either one of the die or both is : 11/36 or about 0.305.

Also I calculate the probability of rolling 4 dice and getting a 6 on either one, two, three or all four dice is : 421/1296 or about 0.32.

Is that correct? I am just surprised to find both the probabilities so close together.

Thank you.

Answer 1

Method #$1$ - split it into disjoint events, and add up their probabilities:

$$\sum\limits_{n=1}^{4}\binom4n\cdot\left(\frac16\right)^{n}\cdot\left(1-\frac16\right)^{4-n}$$

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Method #$2$ - calculate $1$ minus the probability of the complementary event:

$$1-\left(1-\frac16\right)^{4}$$

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The result in both cases is: $$\frac{671}{1296}$$

Answer 2

The probability of getting at least $1$ six when rolling four dice is

$$ 1-(5/6)^{4}=1-625/1296=671/1296\approx 0.518 $$