Question 1025823: In problem 2–42, suppose that American investment institutions enter this
new market, and that their probabilities for successful instruments are
Goldman Sachs 70%
Salomon Brothers 82%
Fidelity 80%
Smith Barney 90%
What is the probability that at least one of these four instruments is successful?
Assume independence.
Answer by mathmate(429)   (Show Source):
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Question:
In problem 2–42, suppose that American investment institutions enter this
new market, and that their probabilities for successful instruments are
Goldman Sachs 70%
Salomon Brothers 82%
Fidelity 80%
Smith Barney 90%
What is the probability that at least one of these four instruments is successful?
Assume independence.
Solution:
Here we have 4 (assumed) independent events each with a given probability.
The rule of multiplication says that the probability of all events will succeed is the product of the respective probabilities.
Since we are looking for at P(A)=P("least one of them will succeed"), which is the complement of P(B)=P("all of them will fail"), we only need to find P(B) and obtain P(A) by the relation P(A)=1-P(B) whenever A and B are complementary.
To find P(B), we need to define the events
P(G)=1-0.70=0.30 that Goldman Sachs will fail
P(S)=1-0.82=0.18 that Salomon Brothers will fail
P(F)=1-0.80=0.20 that Fidelity will fail, and
P(B)=1-0.90=0.10 that Smith Barney will fail.
The probability that ALL of them will fail is therefore given by the multiplication rule as
P(B)=P(fail)=P(G)*P(S)*P(F)*P(B),
and the probability that at least one of the will succeed is
P(A)=1-P(B), as explained above.
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