"If you roll five dice, what are the..." was answered by our experts

prydlyfrl7w

Answered question

2023-01-07

If you roll five dice, what are the odds of rolling five 5's?

Answer & Explanation

tjaran12f

Beginner2023-01-08Added 13 answers

1 Dice: chance of 1 in 6,

\to \frac{1}{6} .

2 dice: chance of 1 in 6 for each, so:

\to \frac{1}{6} . \frac{1}{6} = \frac{1}{36}

For 2 dice: 1 in

6^{2}

= 1 in 36;
Then for 5 dice: 1 in

6^{5}

= 1 in 776 ...

Jasmine Young

Beginner2023-01-09Added 2 answers

The question can be rephrased as, "If we roll one die five times, what is the probability of getting a 5 on each roll?" assuming that the outcomes of all five dice are independent.
Since there is one "successful" outcome (rolling a 5) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6), the probability of rolling a 5 on one roll is

P (roll one 5) = \frac{1}{6} .

Here's where the independence comes in. Since we're treating the 5 dice as independent, the probability of rolling five 5's is the same as the probability of rolling one 5, five times. As such, we can multiply

P (roll one 5)

by itself 5 times, and it will give us the same answer as

P (roll five 5's) .

P (roll five 5's) = {[P (roll one 5)]}^{5}

P (roll five 5's) = {(\frac{1}{6})}^{5}

P (roll five 5's) = \frac{1}{7776}

Finally, since this question asked for the odds of rolling five 5's (and not the probability of it), we take the ratio

\frac{P (rolling five 5's)}{P (NOT rolling five 5's)}

which gives

\frac{1 / 7776}{7775 / 7776} = \frac{1}{7775}

Since this is an odds value, it is often written as

1 : 7775

. This says that for every 1 success, we should expect 7775 failures.

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