Question

The probability that a contractor will get a plumbing contract is 2/3 and an electric contract is 4/9. If the probability of getting at least one contract is 4/5, find that he will get both contracts.​

Answer 1

Let,

• P = event of getting a plumbing contract

• E = event of getting an electric contract

The probability of getting a plumbing contract,

• $P(P)=\dfrac{2}{3}$

The probability of getting an electric contract,

• $P(E)=\dfrac{4}{9}$

The probability of getting at least one contract,

• $P(P\cup E)=\dfrac{4}{5}$

The probability of getting both contracts is given by $P(P\cap E).$

The formula to find $P(P\cap E)$ is,

$\longrightarrow P(P\cap E)=P(P)+P(E)-P(P\cup E)$

Plugging in the values we have,

$\longrightarrow P(P\cap E)=\dfrac{2}{3}+\dfrac{4}{9}-\dfrac{4}{5}$

$\longrightarrow\underline{\underline{P(P\cap E)=\dfrac{14}{45}}}$

Hence the probability of getting both contracts is 14/45.

Answer 2

Given :-

The probability that a contractor will get a plumbing contract is 2/3 and an electric contract is 4/9. If the probability of getting at least one contract is 4/5,

To Find :-

Both contracts

Solution :-

At first

If he is a plumbing contractor

2/3

If he is a electric contractor

4/9

Now

If he is one of them

4/5

So,

Probability of plumbing contractor + probability of electric contractor

²/₃ + ⁴/₉

⁶ ⁺ ⁴/₉

¹⁰/₉

Now,

Probability of both = ¹⁰/₉ - ⁴/₅

= ⁵⁰ ⁻ ³⁶/₄₅

= ¹⁴/₄₅