What are the odds of rolling at least one “6” when throwing 6 dice.

Question

I swear I leaned this in school at some point but its been bothering me .

If you throw a six sided die, the odds of rolling a six are 1/6.

But what are the odds of rolling at least one six if you throw two dice at the same time? I thought it was 1/6 + 1/6 or 2/6, but suspect I might be wrong.

This is the idea that has me stuck. If you throw 6 dice at the same time what are the odds of rolling at least one 6 ? It can’t be 6/6 or 100%. The odds of rolling at least one 6 are certainly high but absolutely not 100%, so the logic I used for the two dice can’t be correct. There must be a formula for this but I’m having trouble searching for it . Thanks !

In: 12

Answer 1

Let’s think of odds as a portion of all the possible outcomes. For this example lets use coins because manually listing the possible outcomes of dice is a PITA.

Flipping a coin 4 times, every possibility is: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT

We can calculate the total amount of outcomes by taking the amount of possible outcomes of a single “turn,” (in this case 2) and taking it to the power of the amount of “turns,” we have (in this case 4). 2^(4) = 16 possible outcomes.

Which odds we get depends on which questions we ask.

**What are the odds we get at least 3 tails?** Well, we count up the amount of outcomes that have three tails in them (4) and take that out of the total outcomes (16). 4/16 or simplified down we have a 1/4 chance of getting at least 3 tails.

**What are the odds we get 3 tails in a row?** There are only 3 outcomes that have 3 in a row, meaning our odds are 3/16.

**What are the odds we get the same number of heads as tails?** 6 outcomes have this property, so our odds are 6/16 or 3/8.

You can do the same thing with dice. If we listed out all the possible outcomes of rolling a dice twice (same as rolling 2 dice at once), with each chance having 6 possible outcomes. 6^(2) = 36 possible outcomes total. Each roll has a possibility of landing on a 6 once, which means 2 out of the 36 possibilities, or in other words we have a 2/36 = 1/13 chance or rolling a 6 at least once.

Give it a shot yourself: What do you think the odds will be of getting both dice to land on a 6?