A Probability Question Featuring A Six Sided Die

Hi there. This is a short probability post that is based on a question I received from a tutoring client. I thought my probability skills have went down over the years but I did manage to get this one right fairly fast.

Screenshots are from Desmos. Math images are done in LaTeX and with Quicklatex.com.

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Six Sided Die Question

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You roll a standard and fair six sided die six times. What is the probability that you get all 6 unique numbers (1, 2, 3, 4, 5, 6) in all the six rolls?

In the first roll out of six you get your first unique number. This probability would be 1 or (6/6) or six out of six.

With the second roll you cannot roll the number you rolled before. No duplicates allowed. The probability of obtaining the second unique number out of six would be (5/6). It is because there are 5 choices for the second unique number out of 6 possible numbers on a six sided die.

Roll number three would have a probability of (4/6).

The fourth roll would have a probability of (3/6) for obtaining the fourth unique number.

Roll number five would have a probability of (2/6) for obtaining the fifth unique number.

In the sixth and final roll there is one more unique number to obtain. This last probability is (1/6).

Combine these probabilities from each of the six rolls through multiplication. Independence in the probability context is assumed.

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You Can Use Factorials For The Answer

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You can also write the answer to this question with the use of factorials. Instead of writing 6 x 5 x 4 x 3 x 2 x 1 in the numerator you can use 6! to represent that. It looks like a 6 with and exclamation mark. It is more compact to write it this way.

Desmos screenshot below is shown to show other ways to express the answer. You still get 0.01543 or 1.543%.

 

Thank you for reading.

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