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 = \(\frac{17}{105}\)
