# A fair six sided die is rolled four times in a row. What is the probability that die will come up six at least once?

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Sep 21, 2016

$1 - \frac{625}{1296}$

=$\frac{671}{1296}$

#### Explanation:

To get a 6 "at least once", means it can be

$\rightarrow$ Once in 4 rolls,or
$\rightarrow$ Twice in 4 rolls, or
$\rightarrow$ Three times in 4 rolls, or
$\rightarrow$ Four times in 4 rolls

There are obviously many combinations in each of these outcomes which will require lengthy calculations!

Note that the only outcome which is NOT included is

$\rightarrow$ No 6 in 4 rolls.

There is only ONE combination of this happening.

Not 6 and Not 6 and Not 6 and Not 6

$P \left(6\right) = \frac{1}{6} \mathmr{and} P \left(\text{not 6}\right) = \frac{5}{6}$

$P \left(N , N , N , N\right) = \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} = {\left(\frac{5}{6}\right)}^{4}$

$\frac{625}{1296}$

The sum of all the probabilities is always equal to 1.

$\left(\text{Prob of at least one 6") = 1 - P("no 6}\right)$

$1 - \frac{625}{1296}$

=$\frac{671}{1296}$

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#### Explanation

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Jul 5, 2016

$\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$ or $4 \left(\frac{1}{6}\right) = \frac{4}{6} = \frac{2}{3}$

#### Explanation:

Each time we roll a fair six-sided die, there is a 1 in 6 chance that it will come up as a six. We can then use this to figure out what the chance is that a six will be rolled at least once over 4 throws. Because there is often confusion about whether to use addition or multiplication, let's do the math two different ways and see what happens:

The first way is to take each throw and see it as an individual event, each one having a 1 in 6 chance of getting a six, and that there are four of them:

$\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$

The other way to view it is to say that there are four throws, each with a chance one in six chance of being a six:

$4 \left(\frac{1}{6}\right) = \frac{4}{6} = \frac{2}{3}$

How you think about the throws will determine how you do the math.

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