# What is the probability that a triple will occur within the first five rolls of three dice?

kjcdb8er | Teacher | (Level 1) Associate Educator

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Each die roll is independent of the other. This means you can consider each event separately. So instead of thinking about it like you throw all the dice at once, toss one at a time. Throw the first die: what are the odds that it will be a number between 1 and 6? ( 100% ) Let the die roll be N.

Now, what are the odds that the second die will also be N? ( 1/6 )

Now, what are the odds that the third die will be N? (also 1/6)

Multiply the three independent probabilities together to get the probability of the triple occurring: 1 x 1/6 x 1/6 = 1/36 = 0.028

Now, the dice have no memory. Each time you roll the dice, the odds of a triple is 1/36. So the odds that you did NOT roll a triple are 35/36.

Now, what are the odds that you don't roll a triple 5 times? Each time the probability is 35/36, so the probability is (35/36)^5 = 0.8686

The inverse of you did NOT roll the triple is that you DID roll the triple; so the odds of that are 1-0.8686 = 0.1314

So, you have a 13% chance of rolling a triple within 5 rolls.