Jack rolls 5 fair six-sided dice. What is the probability that at least three dice show the same number?
+128474 P ( 3 show the same number ) = C(5,3) (1/6)^3 ( 5/6) (4/6) = 25/ 972 = 200/7776
P( 4 show the same number) = C(5,4) (1/6)^4 ( 5/6) = 25/7776
P( 5 show the same number ) = C(5,5) ( 1/6)^5 = 1/7776
Total probability = (200 + 25 + 1) / 7776 = 226 /7776 = 113/ 3888
+2666 CPhill, your answer is incorrect.
The probability that exactly 3 dice show the same is \({5 \choose 3} \times {1 \over 6^3} \times{5 \over6^2} \times 6\). \(5 \choose 3\) ways to pick the rolls that are the same, \({1 \over 6}^3\) chance of 3 successes, \({5 \over 6}^2\) chance of 2 failures, and 6 different trios of numbers that are the same.
The probability that exactly 4 dice show the same is \({5 \choose 4} \times {1 \over6}^4\times {5 \over6} \times 6\). With the same logic as above.
The probability that all 5 dice show the same is \({1 \over 6}^5\times6\).
Add everything up, and you get \(\color{brown}\boxed{23\over108}\)
