I roll 5 six sided dice simultaneously.
What is the probability that precisely three dice show the same number?
My thoughts:
There are 6^5 = 7776 outcomes.
5C3 ways in which 3 dice show the same number = 10.
However, 10/7776 is not the correct probability.
I feel like I need to account for the number of dice that are not all three however I am unsure how to do this.
Thank you in advance.
Best Answer
The probability doesn't depend on whether you throw the dice simultaneously.
It's not clear from your question whether you want exactly three or at least three identical numbers.
The probability for $5$ identical numbers is $\binom55\cdot6\cdot6^{-5}=6^{-4}$, since we can choose the $5$ dice in $\binom55$ ways and choose $1$ of $6$ numbers for them.
The probability for exactly $4$ identical numbers is $\binom54\cdot6\cdot5\cdot6^{-5}=25\cdot6^{-4}$, since we can choose the $4$ dice in $\binom54$ ways, choose $1$ of $6$ numbers for them and choose $1$ of the remaining $5$ numbers for the remaining die.
The probability for exactly $3$ identical numbers is $\binom53\cdot6\cdot5^2\cdot6^{-5}=250\cdot6^{-4}$, since we can choose the $3$ dice in $\binom53$ ways, choose $1$ of $6$ numbers for them and choose $1$ of the remaining $5$ numbers for each of the two remaining dice.
The probability for at least $3$ identical numbers is therefore $(250+25+1)\cdot6^{-4}=\frac{23}{108}\approx0.213$.