if you were to roll a four-sided die 100 times what is the probability that you will not roll a four?

Question

I was playing d&d and using a online roller that has the option to pick how many times you rolled the die, so out of curiosity I set it to the max (100) and pressed roll about twenty times and I always got at least 20 fours so I tried four a little while and I could not figure it out so I looked it up and came here.

Answer 1

For each roll the probability to not get a 4 is $\frac34$ then for 100 rolls we have

$$p=\left(\frac34\right)^{100}\approx 3.21\cdot10^{-13}\approx 3.21\cdot10^{-11}\%$$

Answer 2

If you roll a d$4$ once, the probability that you don't roll a $4$ is $3/4$. If you roll it $100$ times, the probability that you don't roll a $4$ at any point is

$$ \left(\frac34\right)^{100} \approx 0.00000000000032072 $$

Answer 3

You must roll a $1,2,3$ with probability $\frac{3}{4}$, $100$ consecutive times. By independence and multiplicity, we have

$$p=\left(\frac{3}{4}\right)^{100}\approx 3.207\cdot10^{-13}$$

Can you use this to find the probability of getting at least one four in $100$ tosses?