Question

three rotten apples are mixed with seven good ones and five rotten oranges are mixed with ten good ones. Two fruits are drawn at random. what is the probability that either both are rotten or both are oranges

Answer 1

Dear student,
we are given 7 good apples and 3 rotten apples, so total apples = 10
10 good oranges and 5 rotten oranges, so total oranges = 15
total fruits = 15+10=25

Now,let A be the event of getting an orange
$P\left(A\right)=\frac{{}^{15}C_2}{{}^{25}C_2}$
let B be the event of getting a rotten fruit
$P\left(B\right)=\frac{{}^{8}C_2}{{}^{25}C_2}$
$Now,P(A\cap B)=\frac{5}{25}$
therefore, probability that either both fruits chosen are rotten or both are oranges will be:
$$\begin{gathered} P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B) \\ =\frac{{}^{15}C_2}{{}^{25}C_2}+\frac{{}^{8}C_2}{{}^{25}C_2}-\frac{5}{25} \end{gathered}$$