GRE Probability: A Roll of the Dice
April 9, 2012
Each of 3 fair dice has sides numbered 1, 2, 3, 4, 5, and 6. If these 3 dice are all rolled at the same time, what is the probability that exactly 2 of these dice will show a 1?
For any one die, there is a 1/6 probability of rolling a 1. That also tells us that there is a 5/6 probability of rolling any other number.
Since we’re rolling 3 dice at once, and we want the probability that exactly 2 of them will show a 1, we can quickly figure out how many ways that can happen by jotting some notes down. We could roll 1 – 1 – x, or 1 – x – 1, or x – 1 – 1 (where x indicates any other number than 1.)
Any one of those outcomes represents exactly 2 dice showing a 1; therefore, there are 3 desired outcomes.
In order to determine the probability of one toss of the dice, we multiply the probability of the outcomes together. That means we have to multiply the probability of rolling a 1, times the probability or rolling a second 1, times the probability of rolling any other number:
This is where you have to really pay attention. One of the answer choices will very likely be 5/216.
However, that will not be the correct answer choice! You see, we just calculated the probability of rolling 1, 1, x. BUT, when we jotted down all of the outcomes that would give us exactly 2 dice showing a 1, there were two other possible outcomes. That means that we need to take the probability of 1, 1, x and multiply it times 3 to account for the probability of all three possible outcomes:
One of the keys to probability problems on the GRE is paying attention to exactly what you are solving for, and to what it is that you know at any given moment. Remember to always read the question one more time to make sure that the answer you select truly does answer the question at hand!
If you have questions about probability, ask them here.