Chapter 14 - Probability

If $A, B, C$ are three events associated with a random experiment, prove that:
$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)$.

A bag contains $8$ red and $5$ white balls. Three balls are drawn at random. Find the probability that: (i) All three balls are red, (ii) Exactly one ball is red, (iii) At least two balls are red.

In a relay race, there are $5$ participating teams $A, B, C, D,$ and $E$. (i) What is the probability that $A, B,$ and $C$ finish first, second, and third, respectively? (ii) What is the probability that $A, B,$ and $C$ finish in the first three positions in any order?

Two students $X$ and $Y$ appear in an examination. The probability that $X$ qualifies the exam is $0.05$ and that $Y$ qualifies is $0.10$. The probability that both qualify is $0.02$. Find the probability that: (i) Both will not qualify, (ii) At least one of them will not qualify, (iii) Only one of them will qualify.

Find the probability that when a hand of $7$ cards is drawn from a well-shuffled pack of $52$ cards, it contains: (i) all $4$ Kings, (ii) exactly $3$ Kings, (iii) at least $3$ Kings.

If $P(A) = x, P(B) = y,$ and $P(A \cap B) = z$, express the following probabilities in terms of $x, y, z$:
(i) $P(\text{neither } A \text{ nor } B)$,
(ii) $P(\text{exactly one of } A, B)$,
(iii) $P(A \cap B')$.

Four cards are drawn from a well-shuffled deck of $52$ cards. What is the probability of obtaining $2$ red cards and $2$ black cards?

A box contains $100$ tickets numbered $1, 2, \dots, 100$. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $10$. Find the probability that the minimum number on them is $5$.