Rolling a dice 15 times, find the probability of 6 consecutive rolls have 6 distinct numbers....

Rolling a dice 15 times, find the probability of 6 consecutive rolls have 6 distinct numbers. (i.e. 423123456..., 45612335626..., etc.)

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Expert Solution

Answer:

A dice is rolled 15 times. We have to find the probability of 6 consecutive rolls have 6 distinct numbers.

The probability = (Number of favorable cases)/(Number of total outcomes)

Here, number of total outcomes = 615 [since, there are 6 different outcomes and each can be repeated upto 15 times, therefore the total number of arrangements is 615]

Now, in order to have 6 distinct numbers in 6 consecutive rolls of the die we consider 6 distinct numbers as a single set of rolls. Within this set, the 6 different numbers can be arranged in 6 ! ways (as repeatation is not allowed).

There are 9 different rolls remaining.

Considering the set of 6 rolls and 9 different rolls, we have total 10 different rolls. Out of these 10 rolls our set can be arranged in 10(1) ways i.e. 10 ways.

For the remaining 9 different rolls, we do not have any restriction on the outcomes and hence the total number of cases = 69 [since, each of the 6 different outcomes can be repeated upto 9 times]

Therefore, the number of favorable cases = 10(1) X (6 !) X 69

Thus, the required probability = (10(1) X (6 !) X 69)/615 = 25/162 = 0.1543

Answer: The probability that 6 consecutive rolls will have 6 distinct numbers is 25/162 or 0.1543.

orchestra answered 11 months ago

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