4 dices are thrown at the same time. What is the probability

Here is my approach:

If we name the dices: 1-2-3-4
there are 6 possibilities that just 2 of them are the same: \(1&2-1&3-1&4-2&3-2&4-3&4\)
and to calculate probability of the first one, 1&2, we know that the first dice can be every thing. but the second should be the same as first. the third should be something else than the first and second, and finally the forth should be something else than the three previous ones. So:
\(1*\frac{1}{6}*\frac{5}{6}*\frac{4}{6}\)

and as i said, we have 6 possibilities. So total probability is:
\(6*1*\frac{1}{6}*\frac{5}{6}*\frac{4}{6}=\frac{5}{9}\)

Did the exact same thing. I hope that having similar thought processes to Bunuel's continues...

Please explain why am i going wrong -

favourable outcomes -
1st dice can have any number..hence, 6 possible ways
2nd dice ( same number ) = 1 way
3rd dice = 5 ways
4th dice = 4 ways
total favourable outcome = 6*1*5*4 = 120 ways

total outcome = 6*6*6*6 ways = 1296.

Hence, probablity = 120/1296 = 5/54

The problem is you assumed that 1st dice AND 2nd dice will be 6 and 1. That's not true.

It could be:
2213
1223
1231

So, you must choose the index of the dices that will have the same sides.

Favorable= 4C2*6*1*5*4

You are missing the number of permutations.
6*1*5*4 is perfect. But take an example the four digits are 1,1,3,4. This can further be arranged in 4!/2! ways.
Hope that clarifies!!

Actually combination... 4C2 = 4! / 2! (4-2)!

Done in the same manner:

Probability of getting two same faces: 1 and 1/6
For other two faces, probabilities: 5/6 and 4/6 respectively.
The dice combinations can be arranged in 4!/2! ways
Total probability = 1*1/6*5/6*4/6*4!/2! = 5/9

\(= (\frac{1}{6}*\frac{1}{6}*6)*\frac{5}{6}*\frac{4}{6}*\frac{4!}{2!*2!}\) \(= \frac{5}{9}\)

we multiply by 6 because there are 6 possibilities of having same numbers {1,1}.. {6,6} etc. we multiply by 4! for possible combinations in between the four dices, by 2! for two same numbers and again by 2! for possibilities between same & not-same numbers.

Say: only 1st and 2nd show the same face
possible outcome 6^4

1st dice = 6c1=6
2nd dice= 1c1 [same as 1st]=1
this can happen in 6 different ways

3rd dice= 5c1=5
4th dice= 4c1=4

(6*1*6*5*4)/6^4=5/9

I also did it this way. Now, the only difference btw Bunuel's method and this one is that we use 6C1 instead of 4C2. Can anybody help me understand if this approach is wrong in any way?

It's not clear what does the red part in the above solution mean.

If it means that there are total of 4C2=6 ways to chose which two show the same face then the logic is correct.

Well I'm not sure about BDSunDevil, but for me the red part means : there are 6 different ways the 1st and 2nd dice can correlate.
i.e
if both D1 and D2 :1
OR if both D1 and D2 :2
and so on till 6.

Let me further elaborate:

1. We know that the both die can have one of the following numbers i.e 1-6.
So possible combinations for D1 and D2: 6C1*1C1 (since D1 and D2 will have to be the same number)

2. Now, the 3rd dice can be any of the remaining 5 numbers. So, possible combinations: 5C1

3. Finally, the 4th dice can be any of the remaining 4 numbers. So, possible combinations: 4C1

From 1,2 and 3: Total Favorable Combinations: 6C1*1C1*5C1*4C1

Total Outcomes(as per your logic): 6^4

So, P(F)= (6C1*1C1*5C1*4C1)/ (6^4)

I have no problems with your logic either, just want to know if this one is possible also.

4 dices are thrown at the same time. whats the probability of getting ONLY 2 dices showing the same face?

I suppose "only 2 dice showing the same face" means EXACTLY two? If so then:

Total # of outcomes = 6^4

Favorable outcomes = 4C2=6, choosing two dice which will provide the same face, these two dice can take 6 values, other two 5 and 4. So, favorable outcomes=4C2*6*5*4.

Answer: 5/9.

************

This was a solution posted by you on probability
can you please explain why is the 6 shown twice ??


why is the 6 given twce it shuld be only ONCE

becuase the only possibility of 2 dices getting the same number is

66
55
44
33
22
11

that is 6 possibilities


so 6*5*4/6^4

why is the 6C2 in the picture

please explain

We have 4 dice (A, B, C, D), not 2. So, \(C^2_4\) is the ways to select which two dice out of 4, will provide the same face: (A, B), (A, C), (A, D), (B, C), (B, D), or (C, D) . Next, these two dice can take 6 values, the remaining two 5 and 4, respectively. So, the # of favorable outcomes is \(C^2_4*6*5*4\).

Hope it's clear.

6C1 refers to the number of possibilities for the two same faced dice can have (1, 2, 3, 4, 5, 6). This is wrong as it was already counted for in the calculation.

4C2 refers to the number of possibilities the four dice can be arranged (1/1/2/3 as opposed to 2/3/1/1).

However, I think the question was not clear as to whether the order of the four dice matters...

Is it just a coincidence that Bunuel's method, with 4C2 = 6, is the same as the # of pairings you had (1&2-1&3...etc). I don't understand why those should be the same #.
4C2 is simply the combinations of 2 dice chosen out of 4. Is yours a long form way of saying the same thing?

1. The formula is\(nCr\)
2. n is 6 as a dice has 6 values
3. r is 1 as only one value is shown
4. But there are 4 dices and there is a constraint on the value of n for 3 dices
5. Let the first dice show any value. Thus it can show 6 values and n is 6. r is 1
6. The second dice in consideration has to show the same value. Therefore n is 1. r is 1
7.The third dice cannot show the value on the first two dices and therefore n is 5. r is 1
8. Similarly for the fourth dice n is 4. r is 1
9. We can select 2 dices out of 4 dices in \(4C2\)ways
10. The total number of ways of exactly 2 dices out of 4 dices showing the same face is \(4C2 *6C1*1C1*5C1*4C1= 720\)
11. The total number of possibilities with any value with 4 dices is 6^4=1296
12. The probability is 720/1296 = 5/9