Dice sequence likelihood

Probability Level 2

Teuta rolled a fair six-sided die ten times and got the following sequence of results:

3 → 6 → 5 → 3 → 6 → 3 → 4 → 3 → 1 → 2 3 \rightarrow 6 \rightarrow 5 \rightarrow 3 \rightarrow 6 \rightarrow 3 \rightarrow 4 \rightarrow 3 \rightarrow 1 \rightarrow 2

She thought this was rather unremarkable. But how does the likelihood of this sequence compare to the likelihood of rolling ten times and getting 6 every time?

6 → 6 → 6 → 6 → 6 → 6 → 6 → 6 → 6 → 6 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6 \rightarrow 6

The sequences are equally likely Ten 6s is less likely Ten 6s is more likely

Edward Teach by Edward Teach , 34, USA

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2 solutions

Blan Morrison
Feb 22, 2018

Every possible sequence of 10 dice rolls has a 1 6 10 \frac{1}{6}^{10} chance of happening. Therefore, both sequences have an equal probability of occurring.

The reason people might get tricked by this problem is because there is a 6 × 1 6 10 = 1 6 9 6\times\frac{1}{6^{10}}=\frac{1}{6}^9 chance of a consecutive string of numbers.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Giorgos K.
Feb 22, 2018

1/6^10 each

which is1/60466176

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