Three dice
Task number: 3537
Determine the probability that in a roll of six dice at least three of the dice will have a value of at least three.
Solution
We analyze cases by the number of dice that have a value of at least three. For \(i\) dice the probability is
\(\displaystyle \binom{6}{i}\left(\frac23\right)^i\left(\frac13\right)^{6-i}\)
Adding the values for \(i=6,…,3\) we get
\(\displaystyle \binom{6}{6}\left(\frac23\right)^6 + \binom{6}{5}\left(\frac23\right)^5\left.\frac13\right.+ \binom{6}{4}\left(\frac23\right)^4\left(\frac13\right)^2+ \binom{6}{3}\left(\frac23\right)^3\left(\frac13\right)^3= \frac{656}{729}\)
Answer
The probability of the given event is \(\frac{656}{729}\doteq 0{,}9\).
