Occupancy Probability Page 8 How would we calculate the probability of getting 2 or more sixes in four rolls of a six-sided die? STEP 1 The number of trials, "n", is 4. The number of successes (or "r") will be 2, 3 or 4. The probability, "p", equals 1/6 or 0.166666666666... Now, we'll calculate the total number of combinations that 4 rolls of a six-sided die can produce: 64 = 1,296 combinations 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   6   N  N 6   N   6  N 6   N   N  6 N   6   6  N N   6   N  6 N   N   6  6 "N" represents the numbers 1 through 5 and "6" just means the number "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: 1 * 1 * 5 * 5 = 25 combinations 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: 6 times 25 or 150 The probability of rolling exactly two sixes in 4 rolls of a die is: 150 / 1,296 = 0.115740740740741 STEP 3 In how many ways can exactly 3 sixes appear in 4 "throws" of a die? The 3 sixes can be placed in the 4 bins in four different combinations with "N" representing the numbers 1 through 5. 6   6   6   N 6   6   N   6 6   N   6   6 N   6   6   6 Since each of the 4 rows of "bins" represents 5 combinations, this makes a total of 20 combinations. The probability of rolling exactly 3 sixes in 4 rolls of a die is: 20 / 1,296 = 0.0154320987654321 STEP 4 The third calculation to make is the probability of getting exactly four sixes in 4 rolls of a die. Here are all the combinations of rolling 4 sixes. 6   6   6   6 So, as can be seen, there is only one combination in which all 4 sixes appear. The probability of exactly 4 sixes occurrring is: 1 / 1,296 = 0.000771604938272 STEP 5 Now we sum the probabilities: 2 sixes = 0.115740740740741 3 sixes = 0.015432098765432 4 sixes = 0.000771604938272 Total = 0.131944444444445 That is the probability of getting at least 2 sixes after rolling a 6-sided die 4 times. 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) 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