eProbability

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. $X_{i} = [\\ mbox{I get 1 out of 6 times}]$ .

If you throw the dice 6 times, there are a total of ${}_6 \Pi_{6} = 6^6$ . 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 ${}_5 \Pi_{5}$ ways． The same can be said for the second throw, the third throw，so the total is

$\displaystyle {}_6 C_{1} \cdot {}_5 \Pi_{5} = 6 \cdot 5^5$

Therefore the 1 appears once in 6 times is

$\displaystyle P_{r}(X_{1}) = \frac{6 \cdot 5^5}{6^6} = \left(\frac{5}{6}\right)^5$

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