What is the probability of rolling a number less than 10?

Answer 1

The answer depends upon what you are rolling.

The probability is calculated as:

number_of_ways_of_success ÷ total_number_of_outcomes

There can be shortcuts to finding the number of ways of a successful outcome (for this problem it is a roll of less than 10), but it may end up at having to list out all the possible outcomes and counting the ones which are a success; in this case it may be less to find if the number of failures is instead calculated and the probability of a failure subtracted from 1 since probability_of_success + probability_of_failure = 1

If you are rolling 2 standard dice, there are 36 possible outcomes. A success is if the sum is less than 10, so a failure is if the sum is 10 or more:

12 - can be made only 1 way: (6, 6)

11 - can be made 2 ways: (5, 6), (6, 5)

10 can be made 3 ways: (4, 6), (5, 5), (6, 4)

→ there are 1+2+3 = 6 ways of failure

→ probability(failure) = pr(roll ≥ 10) = 6/36 = 1/6

→ probability(success) = pr(roll < 10) = 1 - pr(failure) = 1 - pr(roll ≥ 10) = 1 - 1/6 = 5/6.

With an icosahedral die (as used by D&D players) with the numbers 1-20 on it, the probability of rolling less than 10 would be 9/20 as there are 9 numbers (1-9) less than 10 and there are 20 possible outcomes.

Answer 2

If this is with two standard 6 sided dice, then you have 36 outcomes: from 1+1 = 2, up to 6+6=12.

Let's look at the ones which are 10 or greater: 12 (only 1), 11 (there are 2, 5+6 and 6+5), 10 (there are 3: 5+5, 6+4, 4+6), all other rolls will be less than 10.

So we have 6 outcomes which are 10 and up, therefore you have 30 outcomes which are less than 10, so the probability is 30/36 which is 5/6, or about 83%.

Answer 3

On an ordinary number cube it is zero.