Axiomatic Approach to Probability

Three identical dice are rolled. The probability that the same number will appear on each of them is

If the integers $m \& n$ are chosen at random from $1$ to $100$, then the probability that a number of the form $7^m+7^n$ is divisible by $5$ equals:

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, $22240$ is in $X$ while $02244$ and $44422$ are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of $20$ given that it is a multiple of $5$. Then the value of $38 p$ is equal to

Let $A$ and $B$ be two independent events such that $P\left(\overline{A}\right)=0.7, P\left(\overline{B}\right)=k, P\left(A\cup B\right)=0.8$.

What is the value of $k$?

A bag contains $6$ white and $4$ black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is

Let $S=\left\{M=\left[a_{ij}\right], a_{ij}\in \left\{0,1,2\right\}, \left\{1\le i,j\le 2\right\}\right\}$ be a sample space and $A\left\{M\in S:M \text{is invertible}\right\}$ be an even. Then $P\left(A\right)$ is equal to

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q},$ where $p$ and $q$ are co-prime, then $q-p$ is equal to

There are $5$ black and $3$ white balls in a bag. A die is rolled, we need to pick the number of balls appearing on the die. The probability that all the balls are white is:

If three straight lines are drawn on a white board at random with the eyes closed, show that the probability of forming a triangle is $\frac{1}{7}.$

Let $M={\left[a_{ij}\right]}_{2\times 2}, 0\le i, j\le 2$ where $a_{ij}\in \left\{0,1,2\right\}$ and $A$ be the event such that $M$ is invertible then $P\left(A\right)$ is 

Three dice are thrown. The probability that no outcomes are similar is $\frac{p}{q}.$ What is $q-p?$ ($p$ and $q$ are co-primes).

During a social gathering all the friends are following COVID$-19$ norms as well as enjoying the day. They are eating, dancing, playing games. We also became a part of group playing cards, each one started asking the other about the probability of a particular event, let us try to answer some of the event.
In a three card combination what is the probability of a player having two black $7\text{'}s.$

Let $S$ be the sample space and $A\subseteq S$ be an event.

Given below are two statements:

$\left(S_1\right): P\left(A\right)=0$, then $A=\varnothing$.

$\left(S_2\right): P\left(A\right)=1$, then $A=S$.

The probabilities of three events $A,B \& C$ are $P\left(A\right)=0.6, P\left(B\right)=0.4 \text{and} P\left(C\right)=0.5.$ $\text{If} P\left(A\cup B\right)=0.8, P\left(A\cap C\right)=0.3, P\left(A\cap B\cap C\right)=0.2 \& P\left(A\cup B\cup C\right)⩾0.85.$
$\text{Find the range of} P\left(B\cap C\right).$

In a throw of dice the probability of getting one in eleventh throw is$?$

If $\mathrm{P}\left(\mathrm{E}\right)=\frac{2021}{2022}$ then find $\mathrm{P}\left(\overline{\mathrm{E}}\right)$?

A bag contains $15$ red, $15$ green and $15$ blue balls. If the drawn ball is put back, then find the probability of the drawing $2$ green balls.

What is the probability of getting two head if you toss five coins simultaneously?

A bag contains $8$ white and $4$ black balls. Balls are drawn one by one without replacement till all the black balls are drawn. The probability that the procedure of drawing comes to an end at the $6^{\mathrm{th}}$ ball is?