A lot contains $$20$$ articles. The probability that the lot contains exactly $$2$$ defective articles is $$0.4$$ and the probability that the lot contains exactly $$3$$ defective articles is $$0.6$$. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelth testing.

In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random, If he is allowed upto three chances to answer the questions, find the probability that he will get marks in the questions.

In a certain city only two newspapers $$A$$ and $$B$$ are published, it is known that $$25$$% of the city population reads $$A$$ and $$20$$% reads $$B$$ while $$8$$% reads both $$A$$ and $$B$$. It is also known that $$30$$% of those who read $$A$$ but not $$B$$ look into advertisements and $$40$$% of those who read $$B$$ but not $$A$$ look into advertisements while $$50$$% of those who read both $$A$$ and $$B$$ look into advertisements. What is the percentage of the population that reads an advertisement?

$$A, B, C$$ are events such that
$$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$$
$$P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$$

If $$P\left( {A \cup B \cup C} \right) \ge 0.75,$$ then show that $$P$$ $$(BC)$$ lies in the interval $$0.23 \le x \le 0.48$$