Chapter 14 - Probability

A fair die is rolled. Describe the following events: (i) $A$: getting a prime number, (ii) $B$: getting a number greater than $3$, (iii) $A \cap B$, (iv) $A \cup B$. Find their respective probabilities.

A coin is tossed twice. If the second throw results in a tail, a die is thrown. Write the sample space of this experiment.

If $P(A) = \frac{2}{11}$, $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{6}{11}$, find: (i) $P(A \cap B)$, (ii) $P(A \cap B')$, (iii) $P(A' \cap B')$.

Two dice are thrown. Find the probability of getting a total score of at least $10$.

Four cards are drawn at random from a well-shuffled pack of $52$ cards. What is the probability of getting $3$ diamonds and one spade?

A box contains $10$ red, $20$ blue, and $30$ green marbles. Five marbles are drawn from the box. What is the probability that: (i) all will be blue? (ii) at least one will be green?

If $A, B, C$ are three mutually exclusive and exhaustive events associated with an experiment, find $P(A)$ given that $P(B) = \frac{3}{2}P(A)$ and $P(C) = \frac{1}{2}P(B)$.

In a class of $60$ students, $30$ opted for NCC, $32$ opted for NSS and $24$ opted for both. If one student is selected at random, find the probability that: (i) The student opted for NCC or NSS, (ii) The student opted for neither NCC nor NSS.

If the letters of the word 'ALGORITHM' are arranged at random in a row, what is the probability that the letters $G, O, R$ must remain together in that order?

An urn contains $6$ red and $4$ black balls. Two balls are drawn without replacement. Find the probability that both balls are red.

Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined: $P(A) = 0.5, P(B) = 0.7, P(A \cap B) = 0.6$.

A committee of two persons is selected from two men and two women. What is the probability that the committee will have: (i) no man? (ii) one man? (iii) two men?

If $E$ and $F$ are events such that $P(E) = \frac{1}{4}, P(F) = \frac{1}{2}$ and $P(E \cap F) = \frac{1}{8}$, find: (i) $P(E \cup F)$, (ii) $P(\text{not } E \text{ and not } F)$.

A number is chosen at random from the numbers $1$ to $50$. Find the probability that it is divisible by $4$ or $6$.

Three coins are tossed. Let $A$ be the event 'three heads appear', $B$ be 'two heads and one tail appear', $C$ be 'three tails appear', and $D$ be 'a head appears on the first coin'. Are $A$ and $B$ mutually exclusive? Are $A$ and $C$ mutually exclusive?