Bonds rated B have a 25% chance of default in five years.

This event has already taken place it is no longer a prediction but we still have to find ways it can happen.
Total bonds = 30% B- rated bonds + 70 C rated bonds = 100
1] B rated bonds
25% chances and we have a total of 30
$$B\ rated\ bonds\ =\ \frac{25}{100\ \ }\cdot\ \frac{30}{1}\ =\ \frac{15}{2}$$

2] C rated bonds
40% chance and we have a total of 70
$$C\ rated\ bonds\ =\ \frac{40}{100\ \ }\cdot\ \frac{70}{1}\ =\ 28$$
$$Total\ =\ \frac{15}{2\ \ }+\ \frac{28}{1}\ $$
Probability that the randomly selected bond is
$$B\ -\ rated\ =\ \frac{15}{2\ \ }\ \frac{ }{ }\ \left[\frac{15}{2}\ +\ \frac{28}{1}\ \right]$$
$$=\ \frac{15}{15\ +\ 56}\ =\ \frac{15}{71\ }$$
Answer is option B

Let total bonds be x
Then B bonds are 0.3x and C bonds are 0.7x
So total chances of default= 0.3x*0.25+ 0.7x*0.4=0.355x
Chances of default of B bond= 0.3x*0.25=0.075x
Probability= 0.075x/0.355x= 15/71

AAPL wrote:Bonds rated B have a 25% chance of default in five years. Bonds rated C have a 40% chance of default in 5 years. A portfolio consists of 30% B-rated bonds and 70% of C-rated bonds. If a randomly selected bond defaults in a five year period, what is the probability that it was a B-rated bond?

A. 3/40
B. 15/71
C. 1/4
D. 3/10
E. 56/71

The probability that a randomly selected bond that defaults in a five year period was a B-rated bond is

(0.3 x 0.25)/(0.3 x 0.25 + 0.7 x 0.4) = (3 x 25)/(3 x 25 + 7 x 40) = 75/355 = 15/71

Answer: B