What is the probability of getting a 3 or higher on a six sided die, if I reroll after failing the first time? Just as the question says... What is the probability of rolling a 3 or higher on a six sided die, if I reroll the die a second time when I fail the first time?
 A: You can do this by considering two cases. Let $A$ be "roll three or higher". Then $$P(A)=P(A\mid E)P(E)+P(A\mid E^c)P(E^c)$$ where $E$ is "failed the first time".
This gives $P(A)= 2/3\cdot 1/3+1\cdot 2/3=8/9$. Another way of seeing this is to calculate the probability of consecutively rolling less than three. This is just $1/3\cdot 1/3=1/9$ so $P(A)=1-1/9=8/9$.
A: Let us pretend that we roll the dice twice and consider the following elementary events $$\{(1f, 2f),(1f, 2s),(1s, 2f),(1s, 2s)\}.$$
(Here the numbers refer to the first and the second trial, and $f$ and $s$ refer to failure and success.) The probabilities of these elementary events are $$\frac 19,\frac29,\frac29,\frac49.$$
We are interested in the following  probability
$$P(\{(1f, 2s),(1s, 2f),(1s, 2s)\})=\frac29+\frac29+\frac49=\frac89.$$
A: The first time you roll, you have probability $\frac{2}{3}$ to succeed. In the case you fail (which happens with probability $\frac{1}{3}$), you roll again, with the same probability of success. So we add $\frac{1}{3}\cdot\frac{2}{3}$. This gives $\frac{2}{3}+\frac{1}{3}\cdot\frac{2}{3}=\frac{8}{9}$.
A: I respectively disagree with the previous answers as each roll of the die is completely independent of the other die. The events are not connected. When I took statistics this was one of the most important concepts that was stressed.
Ie. Flip a coin and if you get 3 heads in a row the next flip you still have a 50/50 chance for heads or tails
So rolling a 3 or higher would mean a 3,4,5 or 6. That is 4 out of 6 for a 2/3 probability or 66%.
