Question

If A and B are two events associated with a random experiment, then
$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

If A, B, C are three events associated with a random experiment, then
$P(A\cup B\cup C)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)$

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective, If a person takes out 2 at random what is the probability that either both are apples or both are good?

Answer 1

$n\left(S\right)={}^{30}{C}_2$

Let A be the event of getting two apples and

B be the event of getting two non-defective fruits.

And $(A\cap B)$ be the event of getting two non-defective apples$(20-5=15non-defectiveapples)$

$P\left(A\right)=\frac{{}^{20}{C}_2}{{}^{30}{C}_2}$

$P\left(B\right)=\frac{{}^{22}{C}_2}{{}^{30}{C}_2}$

And $P(A\cap B)=\frac{{}^{15}{C}_2}{{}^{30}{C}_2}$

$\therefore P(A\cup B)=P\left(A\right)+P\left(B\right)–P(A\cap B)$

$=\frac{{}^{20}{C}_2}{{}^{30}{C}_2}+\frac{{}^{22}{C}_2}{{}^{30}{C}_2}-\frac{{}^{15}{C}_2}{{}^{30}{C}_2}$

$=\frac{526}{435}$