I have two quick questions about probabilities and dice.
- What is the probability that that you can roll AT LEAST two fives in four rolls.
- What is the probability that you roll no more than one five in four rolls.
I'm pretty sure I have been over thinking this, but I just want to check to make sure.
For the first problem the sample set should be ${6^4}$. Then the probability of rolling one 5 should be $\frac{1}{6}$. There are 6 ways to roll two fives. Should the answer to this problem then be $\frac{6}{6^4}$?
The second problem… so the probability of rolling a five is $\frac{1}{6}$ then the probability of not rolling a five would be $\frac{5}{6}$. Therefore, the answer should be $\frac{5^4+4}{6^4}$?
Any help would be much appreciated. Thanks!
Best Answer
Let's model the first problem in the following way; let's check if at each roll 5 has come out or not. The probability of 5 is $1/6$.
Now the problem is asking you at least 2 times in four launches. Because the probability sums to $1$, we can say the solution would be 1 - "the probability of rolling the dice four times and not having any 5" (that would be $(5/6)^4)$ - "the probability of rolling the dice four times and just having one five" $((5/6)^3*(1/6)*4)$
The result is $19/144$ = $13.19$%.
For the second problem you have already the answer, because again we know that probability have to sum to 1. Specifically we have already computed the probability of at least two 5. So the probability of one 5 or less is $1-19/144$ = $86.8$%