Find the probability when we tossing a six side dice six times for following condition ..?

Question

1) the probability of getting same number only once

2)probability of getting same number at least once

Answer 1

1) 6! * (1/6)^6

Answer 2

Very interesting.

1. The probs of getting same number only once.

This mean that, out of the six numbers that came out, there is only one pair of same number, not more.

First, lets figure out the amount of possible positions for the pair.

You see, the pair can be the 1st and 2nd, or 4th and 5th, or 3rd and 5th, and more. You got the idea? Two number, with 6 possible position, where order doesn't matter, it's a combination.

= 6C2

= 6! / ( 4! * 2! )

= ( 6 * 5 ) / ( 2 * 1 )      (we scratch the above 4*3*2*1 with the below 4!)

= 30 / 2

= 15

So, there is 15 possible position for the pair, now we calculate the rest of the number.

For the pair, there is 6 possible number ( one to six ), because there is no more than one pair of same number, the 4 other numbers can't have the same number with the pair or with any other number, so there is only 5*4*3*2 (equal 120) possible way for that 4 numbers value, so, the amount of possible way to have only one pair of number is

= 6 * 120 * 15

= 10800

Total amount of possible way of throwing dice 6 time

= 6 * 6 * 6 * 6 * 6 * 6

= 46656

Probs of getting same number only once

= 10800 / 46656

= 0.2315

= 23.15 %

2. The probs of getting same number at least once.

This is so much easier than the first one. To get at least one same number mean, there can be 2, 3, 4, 5, or six same numbers out of all the numbers. We could calculate it one by one, but there is an easier way. We only need to get the probs of getting no same number from 6 throw, and then subtract one with that probs, and you'll get your answer.

Probs of getting no same number

= 6! / 46656

= 720 / 46656

Probs of getting same number at least once

= 1 - ( 720 / 46656)

= 45936 / 46656

= 0.9846

= 98.46 %

There you go, that was fun. Hope it helps

Answer 3

The other answer is correct, but a bit brief, no ?

In order to have no repeat numbers,

and since the number of possible numbers

is equal to the number of throws,

every number must come up exactly once.

The number of ways that can happen is

equal to the number of permutations of 6 objects,

or 6!

The total number of possible throws is 6^6.

So we have 6! / 6^6

For the second part, it is just the complement of that,

or 1 - 6! / 6^6.

.