What is the probability of getting one pair in yahtzee? A 6-sided die is rolled 5 times. What is the probability of getting one pair? I.e. What is the probability of getting an outcome in the form a,a,b,c,d...For example 4,5,5,2,6 is a an outcome where we got one pair (5's).
 A: There are $6^5=7776$ possible outcomes. 
Among these are ${6\choose 2}\cdot {6\choose 4}\cdot 4!=3600$ with exactly one pair (select the two positions making the pair, select the four values occuring, select an order of these values). Thus the probability of getting exactly one pair (and no triple, no two pairs, no full house, ...) is $\frac{3600}{7776}=\frac{25}{54}$.
A: The probability of a yahtzee is 1/1296. Do you know how to work this out? If so, you do the same analysis to get the desired probability. See this for a detailed explanation.
A: If the dice is fair, the probability to get any of the 6 possible pairs is the same. 
Now think of the following: you need to find the probability of getting 2 same values in any of 5 experiments and then 3 different values out of 5 available. There are 6 ways of achieving this result (since each outcome is equiprobable).
Can you handle it from here?
A: If we first count the number of ways to get a distinct pair (e.g. we roll a $6$ twice and get three different numbers for the other rolls) then we can multiply this by $6$ and divide by $6^5$ to get the desired probability. 
First consider the positions of the two identical rolls. Since we are rolling $5$ times there are $5\choose2$ ways to place these dice. Then since we can only have one pair, there are $5$ outcomes for the first of the remaining dice, $4$ for the second and $3$ for the last. In total there are ${5\choose2}\times5\times4\times3$ ways to get one specific pair. Finally the probability of getting any one pair and only one pair is: 
$$
\frac{6\times{5\choose2}\times5\times4\times3}{6^5}.
$$
