# Probability of a bond

I am dead on this question. I can’t even understand the text :((. How come the first one is the correct answer?

The probability of default for a certain junk bond in a given year is 0.2. What is the probability that zero or one bond of the five defaults in the year ahead in a portfolio of only five of these (Assuming the default risks of these are independent to each other)?

• 0.7373. correct.

• 0.4096.

• 0.0819.

Is this the exact same wording in the text?? If so, it is awfully awkward!

This sounds like a binomial distribution with 5 bonds in total and each has a probability of default = 0.2. What is the probability that 0 or 1 defaults?

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John’s right: it’s a horrible question.

It’s not a junk bond; it’s five different junk bonds, whose probabilities of default are not only equal to each other, they’re statistically independent of each other.

Assuming that,

P\left(0\ or\ 1\ defaults\right) = P\left(0\ defaults\right) + P\left(1\ default\right)
= \binom{5}{0}p^0\left(1 - p\right)^5 + \binom{5}{1}p^1\left(1 - p\right)^4
= \left(1\right)0.2^00.8^5 + \left(5\right)0.2^10.8^4
= \left(1\right)\left(1\right)\left(0.3277\right) + \left(5\right)\left(0.2\right)\left(0.4096\right)
= 0.3277 + 0.4096 = 0.7373
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Thank you so much, both of you.

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My pleasure.