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

Rolling at least one 6, is the exact opposite of rolling exactly zero 6. For the first dice you have 5/6 chance to not roll a 6, AND you need the second dice to not roll a six either, AND the third dice, AND the fourth, and so on

In probability, when you hear AND you should think multiplication. Therefore the answer is ⅚ × ⅚ × …
6 times in total, which is (⅚)^6 ≈ 0.33

This means you have around ~33% chance of rolling zero six when throwing six dices. Therefore in ~67% of the cases, there will not be zero 6 (so either one two three four five or six 6)

And when you hear OR, in probability, you should think +. So for example “What is the probability of hitting 6 OR 1 in 6 throws”, is either 6 OR 1 for the first, AND 6 OR 1 for the second, AND 6 OR 1 for the third, and so on → (⅙+⅙)^6