Embibe Experts Solutions for Exercise 1: Exercise

A fair die is rolled. Consider the events $A=\{1,3,5\},B=\{2,3\}$ and $\mathrm{C}=\{2,3,4,5\}$, find $P(A\cap B),P(A\cup B)$.

In an experiment of drawing a card at random from a pack, the event of getting a spade is denoted by $\mathrm{A}$ and getting a pictured card (King, Queen or Jack) is denoted by $\mathrm{B}$. Find the probabilities of $A,B,A\cap B$ and $A\cup B$.

If $A,B$ are any two events in an experiment and $P\left(B\right)\ne 1$, Show that $P\left(\frac{A}{B^c}\right)=\frac{P\left(A\right)-P(A\cap B)}{1-P\left(B\right)}$.

A fair die is rolled. Consider the events $A=\{1,3,5\},B=\{2,3\}$ and $\mathrm{C}=\{2,3,4,5\}$, find $P(A\mid B),P(B\mid A)$.

If $A, B, C$ are three events in a random experiment, prove the following If $A$ and $B$ are mutually exclusive and$P(A\cup B)\ne 0$, then $\mathrm{P}(\mathrm{A}\mid \mathrm{A}\cup \mathrm{B})=\frac{\mathrm{P}\left(\mathrm{A}\right)}{\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)}$

Suppose that a coin is tossed three times. Let event $A$ be "getting three heads" and $B$ be the event of "getting a head on the first toss". Show that $A$ and $B$ are dependent events.

If one card is drawn at random from a pack of cards then show that the event of getting an ace and getting a heart are independent events.

$A,B$ are two independent events such that, the probability of both the events to occur is $\frac{1}{6}$ and the probability of both the events do not occur is $\frac{1}{3}$, Find $P\left(A\right)$.