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For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc.

 

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Suppose $X$ is the number of successes out of $n$ trials, where the trials are independent of each other. The probability of success of every trial is $p$. The probability that there will be exactly $k$ successes out of $n$ trials is

$$\mathbb{P}(X=k)=\left( \begin{array}{c} n \\ k \end{array} \right) p^k (1-p)^{n-k}, k=0,1,…,n.$$

The expected number of success is $\mathbb{E}(X)=np$.

The probability that an individual will default on his/her credit is $\frac{1}{100}.$ What is the probability that out of $200$ debtors of the bank, there will be at least one credit default in a year. You can assume that whether a given debtor will default or not is independent of the behavior of other debtors.
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