Roll 6 Dice: Solve Probability Puzzle!

[Raybert]

On average, how many times do you need to roll six dice together to see all six different numbers turn up within a single such group roll?

 

[CompuChip]

64.8 times?

 

[regor60]

64.8 times?

I agree, except for the .8 :)

 

[Jonathan Scott]

I agreed with 64.8, for the following reasons:

There are 6! ways of throwing all six values and 6⁶ possible results, so the probability in each throw is 6!/6⁶ = (1*2*3*4*5*6)/(6*6*6*6*6*6) = (4*5)/(6*6*6*6) = 5/(3*3*6*6) = 5/324.

The average interval between such throws (or, as in this case, before the first such throw) is therefore the reciprocal of this, 324/5 = 64.8, as usual for a Poisson distribution.

 

[LudusRex]

I would approach this question by first understanding the concept of probability and the principles behind it. The probability of getting a specific number on a dice roll is 1/6, as there are six possible outcomes (1,2,3,4,5,6) and each has an equal chance of occurring.

Based on this, we can calculate the probability of getting all six numbers in a single roll. This can be done by multiplying the probabilities of getting each number (1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6), which equals to 1/46656. This means that the probability of getting all six numbers in a single roll is very low.

To determine the average number of rolls needed to see all six numbers, we can use the concept of expected value. The expected value is the average outcome of a random event over a large number of trials. In this case, the random event is the roll of six dice.

The formula for expected value is (number of trials * probability of success). In our case, the probability of success is 1/46656 and the number of trials is the number of rolls needed to see all six numbers. So, the formula becomes (number of rolls * 1/46656).

To find the average number of rolls, we need to set the expected value equal to 1, as we want to see all six numbers in a single roll. So, the equation becomes (number of rolls * 1/46656) = 1.

Solving for the number of rolls, we get 46656. This means that on average, we would need to roll six dice 46656 times to see all six numbers turn up within a single group roll.

It is important to note that this is an ideal theoretical scenario and in reality, it may take more or less rolls to see all six numbers. However, as the number of rolls increases, the average number of rolls needed to see all six numbers will approach 46656.