\underline{1}

Example Find the probability that 1 will appear once when you throw one die six times Answer First, let’s find the probability using the number of cases. X_{i} = [ mboxIget1outof6times].

If you throw the dice 6 times, there are a total of _{6}Π_{6} = 6^{6}. Next, let’s count how many times there are cases where the 1 appears once out of 6 times

If you get a 1 on the first throw, the remaining 5 times are other than 1， so _{5}Π_{5} ways． The same can be said for the second throw, the third throw， so the total is

6C_{1} ·_{5}Π_{5} = 6 · 5^{5}

Therefore the 1 appears once in 6 times is

P_{r}(X_{1}) = \frac{6 · 5^{5}}{6^{6}} = (\frac{5}{6})^{5}