## A. Theoretical Review Questions:

**1. Explain the concept of probability and its importance in management.**

**ans:** Probability can be defined as a measure of the likelihood or chance that a particular event will happen.

• It is typically expressed as a number between 0 and 1, where 0 indicates impossibility (the event will not occur), 1 indicates certainty (the event will occur), and values between 0 and 1 represent the likelihood of the event occurring.

**Its importance in management are given below:**

- Decision-Making Under Uncertainty
- Risk Management
- Financial Management
- Performance Evaluation
- Insurance and Risk Assessment

**2. Define events in the theory of probability. What are their types? Explain briefly.**

**ans:** An event is a specific outcome or a collection of outcomes from a random experiment. **For examples**:- Tossing of a coin is a **trail **or random experiment, and getting of a head or tail is an **event**.

**The types of events are:**

- Simple and compound events
- Mutually exclusive events
- Equally likely events
- Exhaustive events
- Favourable events
- Independent and dependent events

**‣ Simple and compound events:**

An event that consists of a single outcome is called **simple event** (or elementary event). **For example**:- rolling a die and getting a 3 is a simple event.

A **compound event** is a combination of two or more simple events. ** For example**:- rolling a die and getting a 3 is a simple event and the event of getting prime numbers (2 or 3 or 5) is a compound event.

• It is also know as composite or mixed event.

**‣ Mutually exclusive events:**

**Mutually exclusive events** are events that cannot occur at the same time. If one event happens, the other cannot. **For instance**, when flipping a coin, the events “getting heads” and “getting tails” are mutually exclusive.

**‣ Equally likely events:**

**Equally likely events** are events where each outcome has the same probability of occurring. **For example**, rolling any number on a fair die is an equally likely event because each number has a probability of 1/6.

**‣ Exhaustive events:**

The total number of all possible outcomes of a random experiment is called the **exhaustive events**.

**For example:-**

- In case of a fair coin, the total number of possible outcomes = 2
- If two coins are tossed simultaneously, then total number of possible outcomes = 4
- In throw of a fair dice, the total number of possible outcomes = 6

**‣ Favourable events:**

**Favourable events** are those outcomes that satisfy a specific condition or criteria. **For example**, in rolling a six-sided die, the event “getting a number greater than 4” has the favourable outcomes 5 and 6.

**‣ Independent and dependent events:**

**Independent events** are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. **For instance**, drawing a red card from a deck and rolling a six on a die are often independent events.

**Dependent events** are events where the occurrence or non-occurrence of one event influences the probability of the other event. **For example**, drawing a red card from a deck of cards and then drawing another red card without replacing the first card are dependent events, because removing the first red card reduces the probability of drawing another red card.

**3. What do you understand by mutually exclusive and independent events? Can two events be mutually exclusive and independent simultaneously?**

**ans:** **Mutually exclusive events** are events that cannot occur at the same time. If one event happens, the other cannot. **For instance**, when flipping a coin, the events “getting heads” and “getting tails” are mutually exclusive.

**Independent events** are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. **For instance**, drawing a red card from a deck and rolling a six on a die are often independent events.

• No, two events cannot be mutually exclusive and independent simultaneously. These two concepts are mutually exclusive themselves, and they represent different relationships between events.

**4. Give classical and statistical definitions of probability and point out their limitations.**

**ans:** The **classical definition of probability** is based on the assumption that all outcomes in a sample space are equally likely. It is expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes.

**Limitations:**

- Assumes equal likelihood, which may not always be realistic in real-world scenarios.
- Applicable mainly to finite sample spaces.

The **statistical definition of probability** is based on the concept of relative frequency. It involves conducting experiments or observations to determine the proportion of times an event occurs in the long run.

**Limitations:**

- Requires a large number of trials for accurate probability estimates, which may not always be feasible.
- Does not provide probabilities for one-time or unique events.

**5. Describe briefly the various schools of thought on probability. Discuss their limitations, if any.**

**ans:** The various schools of thought on probability are:

- Classical or Mathematical
- Relative or Statistical
- Subjective Probability

**6. Write short notes on:**

‣ **Mutually exclusive events and overlapping events (not mutually exclusive events)**

**Mutually exclusive events** are events that cannot occur at the same time. If one event happens, the other cannot. **For instance**, when flipping a coin, the events “getting heads” and “getting tails” are mutually exclusive.

**Overlapping events (or non-mutually exclusive events)** are events that can occur simultaneously. **e.g., **rolling an even number and rolling a prime number on a die, as 2 is both even and prime

‣ **Independent and dependent events**

**Independent events** are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. **For instance**, drawing a red card from a deck and rolling a six on a die are often independent events.

**Dependent events** are events where the occurrence or non-occurrence of one event influences the probability of the other event. **For example**, drawing a red card from a deck of cards and then drawing another red card without replacing the first card are dependent events, because removing the first red card reduces the probability of drawing another red card.

‣ **Mutually exclusive events and Independent events.**

**Mutually exclusive events** are events that cannot occur at the same time. If one event happens, the other cannot. **For instance**, when flipping a coin, the events “getting heads” and “getting tails” are mutually exclusive.

**Independent events** are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. **For instance**, drawing a red card from a deck and rolling a six on a die are often independent events.