Probability Questions and Answers - Problems with the solution

In simple terms, probability refers to the likely hood of occurrence of an event. In a simple random experiment, if there are some favourable outcomes out of the total number of possible outcomes, then the probability of the event can be determined as follows:

$P(E)$ = $\frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$

The total number of possible outcomes is also called the sample space. For example, when we toss a coin, then we can either get a head or a tail. That means the total number of possible outcomes is $2$ (head or tail). When we roll a fair die, the total number of possible outcomes is $6\ (1, 2, 3, 4, 5$ or $6)$. When we toss two coins (or toss one coin twice) then the total number of possible outcomes is $4$, namely, $HH, HT, TH$ and $TT$. Similarly, when we roll two dice at the same time, then the total number of possible outcomes would be $36$ as follows:

$(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)$
$(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)$
$(6,1)$ ………………………………$(6,6)$.

Question:

What is experimental probability?

The probability calculated based on the results of an experiment is called experimental probability. For example suppose we toss a coin $20$ times and we get $11$ heads. Then the experimental probability of heads would be = $\frac{11}{20}$ = $0.55$.
Question:

What is theoretical probability?

Theoretical probability is the likely hood of a particular event based on the number of total possible outcomes. For example, if the experiment is about tossing a coin, then we know that there are two possible outcomes: heads and tails. Thus the probability of heads would be $\frac{1}{2}$ = $0.5$. When the number of trials of an experiment increases, the experimental probability gets closer and closer to the theoretical probability.
Question:

What is cumulative probability?

If an experiment has n possible outcomes, then the total probability of each of these $n$ outcomes is the cumulative probability. The cumulative probability of any experiment is always $1$.
Question:

What is a certain event?

An event whose probability is $1$ is called a certain event. That means, that event is surely going to happen.
Question:

What is an impossible event?

An event whose probability is $0$ is called an impossible event. That means, that event is surely NOT going to happen.
Question:

What is a complementary event and how is its probability calculated?

The complement of an event is another event that includes all the possible outcomes of the sample space that are not there in event $E$. In simpler words, complement of event $E$ is 'not $E$'. It is denoted by $E$'. If the probability of an event $E$ is $P(E)$ then the probability of its complementary event would be: $P(E$') = $1 - P(E)$.
Question:

What is geometric probability?

Probability of events based on geometric figures that are calculated using areas of the geometric figures involved is called geometric probability.

Question:

What is a random experiment?

Solution:

A statistical experiment such that it has more than one possible outcomes and the probability of any of the outcomes cannot be predicted before-hand is called a random experiment. For example, tossing a coin, rolling a die, etc.
Question:

What is sample space of a random experiment?

Solution:

The set of all possible outcomes of a random experiment is called its sample space. For example, in the experiment about tossing a coin the sample space would be {head, tail}.
Question:

What is an event?

Solution:

A subset of the sample space that is of interest to us is called an event. An event consists of all the favourable outcomes of the experiment. For example, if the sample space of rolling a die experiment is $\{1, 2, 3, 4, 5, 6\}$ and we are interested in even number rolled, then our event set would be $\{2, 4, 6\}$.
Question:

Differentiate between simple event and a compound event.

Solution:

If an event is such that only one element of the sample space is a member of the event, then it is called a simple event. If an event is such that more than one element of the sample space form the event, then it is called a compound event.
Question:

What is union and intersection of events?

Solution:

If two events A and B are combined using the conjunction 'or' then it is called the union event of the two events. As in event '$A$ or $B$'. It is denoted by $A \cup B$. If the events $A$ and $B$ are combined using conjunction 'and' then it is called the intersection event of the two events. It is represented as '$A$ and $B$' and denoted by $A \cap B$.
Question:

How do we denote the event "$A$ but not $B$"?

Solution:

This event is denoted by $A - B$ or $A \cap B$'.
Question:

What are mutually exclusive events?

Solution:

If two events $A$ and $B$ are such that they can never occur together, then they are called mutually exclusive events. Thus, $A \cap B$ = $\phi$.

## Word Problems

Example 1:

Sam and Joan are playing a tennis match. If the probability of Sam's win is $0.59$, then find the probability of Joan's win.

Solution:

Let event $A$ = Sam wins and event $B$ = Joan wins. Then,

$P(A)$ = $0.59$

Since if Sam wins, Joan cannot win and if Joan wins, Sam cannot win, so we can say that the events $A$ and $B$ are mutually exclusive. Other than these two events, there are no any other possible outcomes. So,

$P(A) + P(B)$ = $1$

$\therefore\ 0.59 + P(B)$ = $1$

$\therefore\ P(B)$ = $1 - 0.59$ = $0.41\ \leftarrow\ Answer!$
Example 2:

A box contains $4$ red, $5$ blue and $3$ green balls. A ball is drawn at random. Find the probability that the ball is either red or green.

Solution:

Probability of the ball being red = $\frac{4}{12}$

Probability of the ball being green = $\frac{3}{12}$

Therefore probability of either red or green ball = $\frac{4}{2}$$\frac{3}{12}$ = $\frac{7}{12}$ $\leftarrow\ Answer!$