The probability that a contractor will get a plumbing contract is 2/3

Question

The probability that a contractor will get a plumbing contract is 2/3 and the probability that he will get an electric contract is 7/12.If the probability to get
exactly one contract is 5/8,the probability that he does not get any contract is

(A)1/16

(B)1/12

(C)3/32

(D)1/8

(E)1/18

CrackverbalGMAT · Accepted Answer

Let Plumbing Contract = A and Electric Contract = B

We are given P(Plumbing) = P(A) = 2/3 and P(Electrical) = P(B) = 7/12

Also the probability of getting exactly 1 contract, i.e either only A or only B = 5/8
i.e P( only A) + P(only B) = 5/8

We need to find the probability of him getting neither contracts

In Set Theory

The universal set value = 1
P(A U B) + P(A U B)' = 1
Therefore P(A U B)' = 1 - P(A U B)

We have:

P(A U B) = P(A) + P(B) - P(A∩B) 
Therefore P(A∩B) = P(A) + P(B) - P(A U B) ----- (1)

Also P(A U B) = P(only A) + P(Only B) + P(A∩B) ---- (2)

Putting equation (1) in (2), we get
P(A U B) = P(only A) + P(only B) + P(A) + P(B) - P(A U B)

2 * P(A U B) =

P(A U B) =1/2*

P(A U B) = 1/2 * (5/8 + 2/3 + 7/12) = 1/2 * (15/24 + 16/24 + 14/24) = 1/2*45/24 = 15/16

Therefore P(A U B)' = 1 - P(A U B) = 1 - 15/16 = 1/16

Option A

Arun Kumar

nick1816 · Answer

P(A) = \frac{2}{3} = \frac{16}{24}

P(B) = \frac{7}{12 }= \frac{14}{24}

P(A U B) = \frac{1}{2}  = \frac{15}{16}

1- P(A U B) = \frac{1}{16}

Kinshook · Answer

Given: The probability that a contractor will get a plumbing contract is 2/3 and the probability that he will get an electric contract is 7/12. Asked: If the probability to get exactly one contract is 5/8,the probability that he does not get any contract is Screenshot 2020-07-12 at 12.00.15 AM.png 5/8 - y + 5/12 - y = 1/3 2y = 5/8 + 5/12 - 1/3 = 17/24 the probability that he does not get any contract = 5/12 - y = 5/12 - 17/48 = 3/48 = 1/16 IMO A

rishit924 · Answer

Dear Sir,

Can you please elaborate more about the signs (AUB)?
I am not clear what does each sign mean.

Thanks in advance.

twobagels · Answer

There are essentially a system of equations to solve to get the answer.

x=union of probabilities, ie area of overlap of two sets on a 2-set Venn diagram
n= probability of neither

1 =  \frac{5}{8}  + x + n

1=  \frac{2}{3}  +  \frac{7}{12}  - x + n

Adding these two equations yields:  \frac{1}{8} = 2*n

n= \frac{1}{16}

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The probability that a contractor will get a plumbing contract is 2/3

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