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

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Math Expert
Joined: 02 Sep 2009
Posts: 46291
Bonds rated B have a 25% chance of default in five years. Bonds rated [#permalink]

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06 Oct 2017, 06:12
00:00

Difficulty:

95% (hard)

Question Stats:

44% (01:35) correct 56% (01:25) wrong based on 48 sessions

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CHALLENGE QUESTIONS

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

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Math Expert
Joined: 02 Aug 2009
Posts: 5933
Bonds rated B have a 25% chance of default in five years. Bonds rated [#permalink]

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06 Oct 2017, 06:29
2
Bunuel wrote:

CHALLENGE QUESTIONS

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

hi..

In such Qs we have to note that the EVENT has already happened

so first find the ways in which it can happen..

let the total bonds be 100 - 30 B rated and 70 C- rated

1) B rated bonds -
25 % chances and total 30 of them so $$\frac{25}{100}*30=\frac{15}{2}$$
2) C-rated bonds -
40% chances and total 70 of them so $$\frac{40}{100}*70=28$$

so total = $$\frac{15}{2}+28$$

now let's find the prob it is B-rated
= $$\frac{15}{2}/(\frac{15}{2}+28)=\frac{15}{15+56}=\frac{15}{71}$$

ans B
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Manager
Joined: 12 Feb 2017
Posts: 71
Bonds rated B have a 25% chance of default in five years. Bonds rated [#permalink]

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06 Oct 2017, 06:52
1
This sum can be solved using Bayes' Theorem

Lets point out different events.
The event that bond B is selected is say A1.
hence P(A1)= 3/10

The event that bond C is selected is say A2.
hence P(A2)= 7/10

let D be the event that default is selected in five years.

probability of default in bond B= P(D/A1) = 25%= 1/4
probability of default in bond c= P(D/A2) = 40%= 2/5

Now we are asked to find probability of choosing bond B when default is already selected.
i.e. P(A1/D)=?

According to Bayes's Theorem,
P(A1/D)= P(A1)*P(D/A1) / { P(A1)*P(D/A1) + P(A2)*P(D/A2) }

P(A1/D)= 3/10 *(1/4) / { 3/10 *(1/4) + 7/10*(2/5) }
after simplifying above equation ,
P(A1/D) = 15/71

Hope it is a correct approach.

Kudos if it helps.
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Joined: 18 Aug 2016
Posts: 634
GMAT 1: 630 Q47 V29
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Bonds rated B have a 25% chance of default in five years. Bonds rated [#permalink]

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07 Oct 2017, 01:28
1
Bunuel wrote:

CHALLENGE QUESTIONS

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

let the no. of B-bonds in portfolio be 3 and no. of C-bonds in portfolio will be 7

Default-B-bonds is 0.25*3 = 0.75
Default-C-bonds is 0.4*7 = 2.8

B-bond-Default probability = 0.75/(0.75+2.8) = 75/355 = 15/71
B
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Bonds rated B have a 25% chance of default in five years. Bonds rated   [#permalink] 07 Oct 2017, 01:28
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