老师，这道题用C5,2*0.17^2*0.83^3计算的结果是一样的，是巧合吗？可以代替解析里的公式吗？谢谢！

问题如下：

A portfolio manager holds five bonds in a portfolio and each bond has a 1-year default probability of 17%. The event of default for each of the bonds is independent.

What is the probability of exactly two bonds defaulting over the next year?

选项：

A.

1.9%

B.

5.7%

C.

16.5%

D.

32.5%

解释：

中文解析：

带公式计算：

P(K=2) 且 n = 5, p = 0.17.

Entering the variables into the equation, this simplifies to 10 x 0.17^2 x 0.83^3 = 0.1652.

Since the bond defaults are independent and identically distributed Bernoulli random variables, the Binomial distribution can be used to calculate the probability of exactly two bonds defaulting.

The correct formula to use is:

where n is the number of bonds in the portfolio, p is the probability of default of each individual bond, and K is the number of bond defaults over the next year. Thus, this question requires P(K=2) with n = 5 and p = 0.17.

Entering the variables into the equation, this simplifies to 10 x 0.17^2 x 0.83^3 = 0.1652.