- What is Probability?
- Types of Probability
- Theoretical Probability
- Experimental Probability
- Subjective Probability
- Terminology Used in Theoretical Probability
- Event in Probability
- Probability Formulas
- Process of Calculating Probability
- Probability Tree
- Solved Examples
- Practice Problems
- Frequently Asked Questions

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, expressed as a number between `0` and `1`. A probability of `0` indicates impossibility, while a probability of `1` indicates certainty. For example, when flipping a fair coin, the probability of landing heads is `0.5`, as there are only two possible outcomes, heads or tails, and assuming the coin is fair, both outcomes are equally likely. Probability theory is widely used in various fields such as statistics, gambling, and risk assessment, to make informed decisions based on uncertain events.

- Theoretical probability
- Experimental probability
- Subjective probability

Theoretical probability, also known as classical probability, is a concept in mathematics that calculates the likelihood of an event occurring based on theoretical assumptions rather than empirical evidence. It is used when all outcomes of an event are equally likely. In theoretical probability, we rely on the assumption that each outcome has an equal chance of happening. Theoretical probability is often used in situations where we have a clear understanding of all possible outcomes and can make accurate predictions based on that understanding.

**The formula of Theoretical Probability:**

**Example: A box contains `5` red balls, `3` blue balls, and `2` green balls. If one ball is randomly selected from the box, what is the theoretical probability of selecting a blue ball?**

**Solution:**

To find the theoretical probability of selecting a blue ball, we need to determine the ratio of favorable outcomes (blue balls) to the total number of outcomes (all balls in the box).

Total number of balls in the box `= 5` (red) `+` `3` (blue) `+` `2` (green) `= 10` balls

Number of favorable outcomes (blue balls) `= 3`

\(\begin{align*}

\text{The theoretical probability of selecting a blue ball} &= \frac{\text{Favorable outcomes}}{\text{Total outcomes}} \\

&= \frac{3}{10} \\

&= 0.3

\end{align*}\)

So, the theoretical probability of selecting a blue ball from the box is `0.3` or `30%`.

Experimental probability is based on actual outcomes obtained through experimentation or observation. Instead of relying on theoretical calculations, it involves conducting trials or experiments to gather data on the occurrence of an event.

**The formula of Experimental Probability:**

**Example: Suppose you want to find the experimental probability of rolling a `6` on a fair six-sided die. You roll the die `100` times and record the number of times a `6` is rolled. After completing the experiment, you find that a `6` was rolled `18` times out of `100` trials.**

**Solution:**

To calculate the experimental probability of rolling a `6`, divide the number of times a `6` was rolled to the total number of trials:

\(\begin{align*}

\text{Experimental Probability} &= \frac{\text{Number of favorable outcomes}}{\text{Total number of trials}} \\

&= \frac{18}{100} \\

&= 0.18

\end{align*}\)

So, the experimental probability of rolling a `6` on the fair six-sided die is `0.18` or `18%`. This means that based on the experiment results, there is an `18%` chance of rolling a `6` on any roll of the die.

Subjective probability is a type of probability assessment based on personal judgment, beliefs, opinions, or intuition rather than on objective data or mathematical calculations. It reflects an individual's degree of belief or confidence in the likelihood of an event occurring.

**`1`. Sample Space:** The sample space, often denoted by \( S \), is the set of all possible outcomes of a random experiment.

**`2`. Event:** An event is a subset of the sample space, representing one or more outcomes of interest.

**`3`. Outcome:** An outcome is a particular result of a random experiment.

**`4`. Favorable Outcomes:** Favorable outcomes are the outcomes that match the event of interest.

**`5`. Total Number of Outcomes:** The total number of outcomes refers to the count of all possible results in the sample space.

**`6`. Probability:** Probability is a numerical measure of the likelihood of an event occurring, expressed as a value between `0` and `1`, where `0` indicates impossibility and `1` indicates certainty.

**`7`. Theoretical Probability:** Theoretical probability is the likelihood of an event occurring based on mathematical analysis and the assumptions of equally likely outcomes.

**`8`. Experimental Probability:** Experimental probability is the probability of an event occurring based on observed data or experimentation.

**`9`. Independent Events:** Independent events are events where the occurrence of one event does not affect the probability of the other event occurring.

**`10`. Dependent Events:** Dependent events are events where the occurrence of one event affects the probability of the other event occurring.

**`11`. Complementary Events:** Complementary events are two events where one event happening implies the other event not happening, and vice versa.

**`12`. Mutually Exclusive Events:** Mutually exclusive events are events that cannot occur simultaneously.

**`13`. Probability Distribution:** Probability distribution describes the likelihood of each possible outcome of a random variable.

**`14`. Expected Value:** The expected value is the average value of a random variable, calculated by weighting each possible outcome by its probability.

**`15`. Venn Diagram:** A Venn diagram is a visual representation used to illustrate relationships between different sets and events in probability theory.

In probability theory, an event refers to a specific outcome or a set of outcomes of interest that can occur in the context of a random experiment or trial. Events are typically subsets of the sample space, which is the set of all possible outcomes of the experiment.

**For example:** Consider the experiment of rolling a six-sided die. The sample space for this experiment is `{1, 2, 3, 4, 5, 6}`, representing all possible outcomes. Now, an event could be defined as rolling an even number. The event in this case would be the subset `{2, 4, 6}`.

Events can be classified based on various characteristics, such as:

**`1`. Simple Event:** A simple event is an event that consists of a single outcome. For example, rolling a specific number on a die.

**`2`. Compound Event:** A compound event is an event that consists of more than one outcome. For instance, rolling an even number on a die is a compound event.

**`3`. Mutually Exclusive Events:** Mutually exclusive events are events that cannot occur simultaneously. For instance, rolling a `2` and rolling a `4` on a die are mutually exclusive events.

**`4`. Independent Events:** Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. Tossing a coin and rolling a die are two independent events.

Events play a central role in probability calculations, and the probability of an event is a measure of the likelihood of that event occurring. The notation **`P(A)`** is used to represent the **probability of event `A`**.

**`1`. Theoretical Probability Formula:**

\[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]

Theoretical probability calculates the likelihood of an event occurring based on mathematical analysis and the assumption of equally likely outcomes. It involves dividing the number of favorable outcomes (outcomes that match the event of interest) by the total number of outcomes in the sample space.

**`2`. Experimental Probability Formula:**

\[ P(A) = \frac{\text{Number of Times the Event Occurred}}{\text{Total Number of Trials}} \]

Experimental probability determines the likelihood of an event occurring based on observed outcomes from a series of trials or experiments. It involves dividing the number of times the event of interest occurred by the total number of trials conducted.

**`3`. Conditional Probability Formula:**

\[ P(A|B) = \frac{{P(A \cap B)}}{{P(B)}} \]

Conditional probability calculates the likelihood of event `A` occurring given that event `B` has already occurred. It involves dividing the probability of both events `A` and `B` occurring (the intersection of `A` and `B`) by the probability of event `B`.

**`4`. Multiplication Rule for Independent Events:**

\[P(A \cap B) = P(A) \times P(B) \]

The multiplication rule for independent events calculates the probability of both events `A` and `B` occurring when the events are independent. It involves multiplying the probabilities of each event individually.

**`5`. Addition Rule for Mutually Exclusive Events:**

\[ P(A \cup B) = P(A) + P(B) \]

The addition rule for mutually exclusive events calculates the probability of either event `A` or event `B` (or both) occurring when the events are mutually exclusive. It involves adding the probabilities of each event individually.

Step `1`. **Define Event:** Clearly define the event you're interested in, specifying the outcome(s) you want to find the likelihood of.

Step `2`. **Identify Sample Space:** Identify all possible outcomes of the experiment, forming the sample space.

Step `3`. **Count Favorable Outcomes:** Determine how many outcomes correspond to the defined event.

Step `4`. **Count Total Outcomes:** Find the total number of outcomes in the sample space.

Step `5`. **Apply Formula:** Use the appropriate probability formula: \[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]

Step `6`. **Calculate Probability:** Plug the numbers into the formula and perform the calculation.

Step `7`. **Interpret Results:** Interpret the calculated probability in the context of the problem to understand the likelihood of the event occurring.

A probability tree, also known as a decision tree or tree diagram, is a visual tool used in probability theory to represent the outcomes of a series of interconnected events or decisions. It resembles a tree structure, with branches representing possible choices or outcomes at each stage of the process.

Here's how a probability tree typically works:

**`1`. Starting Point:** The probability tree begins with a single node, representing the starting point or initial event of the process.

**`2`. Branches:** From this starting point, branches extend outward, each representing a possible outcome or choice at that stage. These branches are labeled with the possible outcomes or events.

**`3`. Subsequent Events:** For each outcome or choice represented by a branch, additional branches extend from it, representing the possible outcomes of subsequent events or decisions.

**`4`. Probabilities:** Alongside each branch, the probability of the corresponding outcome occurring is often listed. These probabilities can be based on theoretical calculations, observed frequencies, or assumptions.

**`5`. End Points:** The endpoints of the branches represent the outcomes of the process. These outcomes may be individual events or combinations of events.

**Q`1`. In a bag, there are `5` red balls, `3` blue balls, and `2` green balls. If one ball is randomly selected from the bag, what is the probability of selecting a red ball?**

**Solution:**

To find the probability of selecting a red ball, we first need to determine the total number of balls in the bag, which is the sum of red, blue, and green balls.

Total number of balls `= 5` (red) `+ 3` (blue) `+ 2` (green) `= 10` balls.

Next, we count the number of favorable outcomes, which is the number of red balls in the bag.

Number of red balls `= 5`.

Now, we apply the probability formula:

\( P(\text{Red}) = \frac{5}{10} = 0.5 \)

Therefore, the probability of selecting a red ball from the bag is `0.5` or `50%`.

**Q`2`. A standard deck of playing cards contains `52` cards, including `13` hearts. If one card is drawn at random from the deck, what is the probability of selecting a heart?**

**Solution:**

To find the probability of selecting a heart, we first determine the total number of cards in the deck, which is `52`.

Next, we count the number of favorable outcomes, which is the number of heart cards in the deck.

Number of heart cards `= 13`.

Now, we apply the probability formula:

\( P(\text{Heart}) = \frac{13}{52} = \frac{1}{4} = 0.25 \)

Therefore, the probability of selecting a heart card from the deck is \( \frac{1}{4} \) or `0.25`, which is equivalent to `25%`.

**Q`1`. A standard deck of playing cards contains `52` cards, including `4` aces. If one card is drawn at random from the deck, what is the probability of drawing an ace?**

- \( \frac{1}{13} \)
- \( \frac{1}{26} \)
- \( \frac{1}{4} \)
- \( \frac{1}{52} \)

**Answer:** a

**Q`2`. A bag contains `8` red balls, `4` blue balls, and `3` green balls. If one ball is randomly selected from the bag, what is the probability of selecting a blue or green ball?**

- \( \frac{3}{15} \)
- \( \frac{7}{15} \)
- \( \frac{1}{5} \)
- \( \frac{2}{5} \)

**Answer:** b

**Q`3`. In a classroom, `20` students play basketball, `15` students play soccer, and `10` students play both sports. If a student is randomly selected from the classroom, what is the probability that the student plays basketball or soccer?**

- \( \frac{1}{3} \)
- \( \frac{2}{3} \)
- \( \frac{7}{9} \)
- \( \frac{5}{6} \)

**Answer:** c

**Q`4`. A fair six-sided die is rolled. What is the probability of rolling an even number or a number greater than `4`?**

- \( \frac{1}{3} \)
- \( \frac{1}{2} \)
- \( \frac{2}{3} \)
- \( \frac{5}{6} \)

**Answer:** c

**Q`5`. A bag contains `6` red marbles and `4` blue marbles. If two marbles are drawn without replacement, what is the probability that both marbles are red?**

- \( \frac{3}{10} \)
- \( \frac{1}{3} \)
- \( \frac{3}{5} \)
- \( \frac{2}{5} \)

**Answer:** b

**Q`1`. What is probability?**

**Answer:** Probability is a measure of the likelihood of an event occurring. It is expressed as a number between `0` and `1`, where `0` indicates impossibility and `1` indicates certainty.

**Q`2`. What are the different types of probability?**

**Answer:** There are several types of probability, including theoretical probability (based on mathematical analysis), experimental probability (based on observed data), subjective probability (based on personal judgment), conditional probability (probability of an event given another event has already occured), and more.

**Q`3`. How do you calculate probability?**

**Answer:** Probability is calculated by dividing the number of favorable outcomes (outcomes corresponding to the event of interest) by the total number of possible outcomes in the sample space.

**Q`4`. What is the difference between theoretical and experimental probability?**

**Answer:** Theoretical probability is based on mathematical analysis and assumes equally likely outcomes, while experimental probability is based on observed data or experimentation. Theoretical probability provides predictions based on mathematical models, whereas experimental probability relies on actual outcomes from trials or experiments.

**Q`5`. Why is probability important?**

**Answer:** Probability is important because it allows us to quantify uncertainty and make informed decisions in situations involving randomness or uncertainty. It is widely used in various fields such as statistics, gambling, finance, science, and everyday decision making to assess risks, predict outcomes, and analyze data.