How to determine the probability that at least two dice will match when rolled together I'm very unlearned in probability and would appreciate learning how to do this.
I play a silly game where I roll several different dice. There is a d30, d28, d26 and two d24s. I am frequently astounded by how often these dice, when rolled, result in two or more of the five matching in value.
How can I determine the probability that two (or more) dice of different values will match, when rolled together?
I tried modifying a solution to the birthday paradox, but I'm almost certain I did something wrong as it gave me a chance of 54%.
My Solution
A solution to the Birthday Paradox instructed to take the probability that two individuals birthdays do not match ($\frac{364}{365}$) and raise it to the power of the number of possible matches, found with group $k$ by $\frac{(k)(k-1)}{2}$
For my problem, I took the individual chances of each die not matching and multiplied them all together. So:
$\frac{29}{30}*\frac{27}{28}*\frac{25}{26}*\frac{23}{24}^2 = 0.82$
Meaning the chances of a match is roughly 18%, or shy of 1/5.
 A: The probability of not getting any matches is easiest to compute if you imagine rolling the dice with fewest sides first. This ensures that the numbers you've already rolled are always possible outcomes of the next roll, so you don't need to divide into cases depending on that, which could easily get unwieldy.

*

*The first d24 never matches anything, so the probability of surviving that roll is $1$.

*The second d24 rolls different from the single number you have yet with a probability of $\frac{23}{24}$.

*Assuming you've survived the first two rolls, the d26 rolls different from the two different numbers you have yet with a probability of $\frac{24}{26}$.

*The d28 rolls different from the three numbers you have yet with a probability of $\frac{25}{28}$.

*The d30 rolls different from the four numbers you have yet with a probability of $\frac{26}{30}$.

The final probability of getting all-different outcomes is
$$ 1 \cdot \frac{23}{24} \cdot \frac{24}{26} \cdot \frac{25}{28} \cdot \frac{26}{30} \approx 0.68 $$
