The probability that a person will get an electrification contract a (

1.	The probability that a person will get an electrification contract a (2/5) and the probability that he will not get a plumbing contract is (4/7). If the probability of getting at least one contract is (2/3), what is the probability that he will get both?
Answer» Let A denote the event that a person will get electrification contract and B denote the event that the person will get a plumbing contract Given : P(A) = \(\frac{2}{5}\) , P(not B) = P( \(\overline{B}\)) = \(\frac{4}{7}\), P(A or B) = \(\frac{2}{3}\) To find: Probability that he will get both electrification and plumbing contract = P(A and B) Formula used : P(B) = 1 – P( \(\overline{B}\)) P(A or B) = P(A) + P(B) - P(A and B) P(B) = 1 - \(\frac{4}{7}\) = \(\frac{3}{7}\) P(B) = \(\frac{3}{7}\) Probability of getting at least one contract = \(\frac{2}{3}\) \(\frac{2}{3}\) = \(\frac{2}{5}+\frac{3}{7}\) - P(A and B) \(\frac{2}{3}\) = \(\frac{14+15}{35}\) - P(A and B) P(A and B) = \(\frac{29}{35}-\frac{2}{3}\) = \(\frac{87-70}{105}\) = \(\frac{17}{105}\) P(A and B) = \(\frac{17}{105}\) The probability that he will get both electrification and plumbing contract = \(\frac{17}{105}\)

The probability that a person will get an electrification contract a (2/5) and the probability that he will not get a plumbing contract is (4/7). If the probability of getting at least one contract is (2/3), what is the probability that he will get both?

Answer»

Let A denote the event that a person will get electrification contract and B denote the event that the person will get a plumbing contract

Given : P(A) = \(\frac{2}{5}\) , P(not B) = P( \(\overline{B}\)) = \(\frac{4}{7}\), P(A or B) = \(\frac{2}{3}\)

To find: Probability that he will get both electrification and plumbing contract = P(A and B)

Formula used : P(B) = 1 – P( \(\overline{B}\))

P(A or B) = P(A) + P(B) - P(A and B)

P(B) = 1 - \(\frac{4}{7}\) = \(\frac{3}{7}\)

P(B) = \(\frac{3}{7}\)

Probability of getting at least one contract = \(\frac{2}{3}\)

\(\frac{2}{3}\) = \(\frac{2}{5}+\frac{3}{7}\) - P(A and B)

\(\frac{2}{3}\) = \(\frac{14+15}{35}\) - P(A and B)

P(A and B) = \(\frac{29}{35}-\frac{2}{3}\) = \(\frac{87-70}{105}\) = \(\frac{17}{105}\)

P(A and B) = \(\frac{17}{105}\)

The probability that he will get both electrification and plumbing contract = \(\frac{17}{105}\)