How would we calculate the probability of getting 2 or more sixes in four rolls of a six-sided die by using a formula?

STEP 1
First, we'll calculate the number of combinations produced by rolling a die 4 times:

Then, we'll calculate the number of ways we can choose 2, 3 or 4 elements from a set of 4 elements.

A good calcualator for this is located here.

Choosing 2 elements from a set of 4 = 6 ways
Choosing 3 elements from a set of 4 = 4 ways
Choosing 4 elements from a set of 4 = 1 way

STEP 2
We'll use the formula to calculate the probability of rolling exactly 2 sixes in 4 rolls.

Here, the probability ('p') is 1/6 or .166666666, the number of successes ('r') is 2 and the number of attempts or trials ('n') is 4.

Entering this data into the formula:
(n r) = 6   p^r = .166666666²   (1-p)^{(n -r)} =   0.8333333333^{(4 -2)} =
6 • 0.0277777778 • 0.6944444444 =
0.1157407407

STEP 3
Now, we'll calculate the probability of rolling exactly 3 sixes in 4 rolls.

(n r) = 4   p^r = .166666666³   (1-p)^{(n -r)} =   0.8333333333^{(4 -3)} =
4 • 0.0046296296 • 0.8333333333 =

0.0154320988

STEP 4
Now, we'll calculate the probability of rolling exactly 4 sixes in 4 rolls.

(n r) = 1   p^r = .166666666⁴   (1-p)^{(n -r)} =   0.8333333333^{(4 -4)} =
1 • 0.0007716049 • 1 =

0.0007716049

STEP 5

Now we sum the probabilities:

2 sixes = 0.1157407407
3 sixes = 0.0154320988
4 sixes = 0.0007716049

Which totals 0.1319444444061731

That is the probability of getting at least 2 sixes after rolling a 6-sided die 4 times.

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In these next pages, we have calculated the occupancy probabilities of rolling dice that have 4, 6, 8, 12 and 20 sides.
(If you are wondering, these would be dice that are in the shape of the 5 Platonic Solids.)
(Tetrahedron, Hexahedron, Octahedron, Dodecahedron, Icosahedron)

Click here to see the probabilities of a:

4 Sided Die  Probability of all 4 numbers in 7 Rolls

6 sided die  Probability of all 6 numbers in 13 Rolls

8 Sided Die  Probability of all 8 numbers in 20 Rolls

12 Sided Die  Probability of all 12 numbers in 35 Rolls

20 Sided Die  Probability of all 20 numbers in 67 Rolls