How would we calculate the probability of getting 2 or more sixes in four rolls of a sixsided die?
STEP 1
STEP 2 In how many ways can exactly 2 sixes appear in 4 "throws" of a die? Here are all the exactly "2 six" combinations: 6 N 6 N 6 N N 6 N 6 6 N N 6 N 6 N N 6 6 Looking at the first row, we can see that there's only one way to put the sixes in the first and second positions but the third and fourth positions can each be filled with the numbers 1 through 5. Therefore, if we were to calculate the number of combinations for the first row, we would get:
By the same reasoning we could also conclude that the other 5 rows represent 25 combinations. So, the total number of ways that exactly 2 sixes can appear in all of the 1,296 combinations is:
The probability of rolling exactly two sixes in 4 rolls of a die is:
STEP 3
6 6 N 6 6 N 6 6 N 6 6 6
The probability of rolling exactly 3 sixes in 4 rolls of a die is:
STEP 4
The probability of exactly 4 sixes occurrring is:
Now we sum the probabilities:
2 sixes = 0.115740740740741 Total = 0.131944444444445 To see this solved by a formula, click on this link: solving by a formula.
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)
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

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