Module - 1 Probability and Statistics

Lesson - 1 Introduction to Probability for GATE Exam

Probability, often regarded as the mathematics of uncertainty, is a concept that permeates our daily lives and underpins countless decisions we make. Whether we are evaluating the chances of rain on a given day, strategizing in a game of cards, or determining the financial risks of an investment, probability serves as our guiding light through a world filled with unpredictability.

Probability is a measure of how likely something is to happen. In mathematical terms, it's the ratio of the number of favorable outcomes to the total number of possible outcomes in a given situation.

Here, P(E) represents the probability of an event (E). The numerator is the count of outcomes we want, and the denominator is the count of all possible outcomes.

Probability is crucial in decision-making because it helps us:

**Assess Risk**: By calculating probabilities, we can quantify the level of risk associated with different choices or actions. Higher probabilities often mean lower risk.**Make Informed Choices**: When faced with uncertainty, we can use probability to make rational decisions. For example, in gambling or investing, knowing the probability of winning or losing is vital.**Predict Outcomes**: Probability is used in forecasting and prediction. Weather forecasts, for instance, rely on the likelihood of different weather conditions occurring.**Optimize Strategies**: In various fields, including business and engineering, probability helps optimize strategies. For instance, it's used in quality control to determine the likelihood of defects in a production process.**Medical Diagnosis**: In healthcare, probability plays a role in medical diagnosis. Doctors use probabilities to assess the likelihood of a patient having a particular disease based on symptoms and test results.

Probability is a powerful tool that enables us to make better decisions when faced with uncertainty. It provides a way to quantify and analyze the chances of different outcomes, helping us choose the most favorable or risk-averse option.

**Random Experiments**: Random experiments are processes or events in which the outcome is uncertain. These experiments can be physical (like tossing a coin) or conceptual (like selecting a random person from a group).**Outcomes**: Outcomes are the possible results of a random experiment. In a random experiment of rolling a six-sided die, the outcomes are the numbers 1, 2, 3, 4, 5, and 6. These outcomes are mutually exclusive, meaning only one outcome can occur at a time.

**Sample Space (S)**: The sample space is the set that contains all possible outcomes of a random experiment. It's often denoted as*S*. For example, when flipping a coin, the sample space*S*consists of two outcomes: {Heads (H), Tails (T)}.**Events**: An event is a subset of the sample space. It represents specific outcomes or combinations of outcomes. Events can be simple or complex.

sample space example

**Simple Event**: A simple event is an event that consists of a single outcome. For example, when rolling a six-sided die, getting a 4 is a simple event.**Compound Event**: A compound event is an event that consists of multiple outcomes or simple events. For instance, when rolling a die twice and considering the sum of the two rolls, getting a sum of 7 is a compound event because it involves two outcomes (e.g., 3 + 4).**Complementary Event**: The complementary event of an event*A*(denoted as ′_A_′) consists of all outcomes that are not in*A*. For example, if*A*is getting a head (H) when flipping a coin, then ′_A_′ is getting a tail (T).

**Probability**: Probability is a numerical measure of the likelihood or uncertainty associated with an event. It quantifies our belief in the occurrence of an event. Probabilities range from 0 (impossible) to 1 (certain).**Measure of Uncertainty**: Probability serves as a measure of uncertainty because it tells us how uncertain or likely an event is to occur. The closer the probability is to 1, the more certain we are about the event happening, while closer to 0 implies greater uncertainty.

**Classical Probability**: Classical probability is a theoretical approach. It assumes that all outcomes in the sample space are equally likely. For example, when rolling a fair six-sided die, each outcome has a probability of 1/61/6 because there are six equally likely outcomes (1, 2, 3, 4, 5, 6).**Empirical Probability**: Empirical probability is based on observed data or experimentation. It calculates probabilities by counting actual occurrences of events. For instance, the probability of rain tomorrow can be estimated by analyzing historical weather data and counting the number of rainy days.

Understanding these fundamental concepts is crucial for building a solid foundation in probability theory, which becomes increasingly important when dealing with more complex probabilistic scenarios and applications.

**Union (A ∪ B)**: The union of two events, *A* and *B*, denoted as *A_∪_B*, represents the event that at least one of *A* or *B* occurs. In set theory, it combines all elements from both sets.

_A_∪_B_={*x*: *x* ∈ *A* or *x* ∈ *B*}

**Intersection (A ∩ B)**: The intersection of two events, *A* and *B*, denoted as *A_∩_B*, represents the event that both *A* and *B* occur simultaneously.

_A_∩_B_={*x*: *x* ∈ *A* and *x* ∈ *B*}

**Complement (′****A****′)**: The complement of an event *A*, denoted as ′_A_′, represents all outcomes that are not in event *A*.

_A_′={*x*: *x* is not in *A*}

**Probability of an Event (P(A))**: The probability of an event*A*occurring is denoted as*P*(*A*). It represents the likelihood of event*A*happening :*P*(*A*)**Probability of Multiple Events (P(A ∪ B), P(A ∩ B))**: You can use the same notation with set operations to denote probabilities of combined events. For example,*P*(*A_∪_B*) is the probability that at least one of*A*or*B*occurs, and*P*(*A_∩_B*) is the probability that both*A*and*B*occur simultaneously :*P*(*A_∪_B*) and*P*(*A_∩_B*)

**Probability of Complementary Event (P(A'))**: The probability that an event *A* does not occur is denoted as *P*(_A_′). It represents the likelihood of the opposite event happening: *P*(_A_′)

Using set notation and probability notation in this way allows us to describe complex events and their probabilities, making it a powerful tool for analyzing probabilistic situations.

- In a bag, there are 5 red balls and 3 blue balls. Calculate the probability of drawing a red ball randomly.

**Answer**

The probability of drawing a red ball can be calculated using the formula:

Probability of Red = (Number of Red Balls) / (Total Number of Balls)

Where the number of red balls is 5, and the total number of balls is 8 (5 red + 3 blue).

Probability of Red = 5 / 8 = 0.625

So, the probability of drawing a red ball randomly is 0.625 or 62.5%.

The first axiom of probability states that the probability of any event, denoted as *P*(*E*), is always a non-negative number. In other words, the probability of any event occurring cannot be negative. It ensures that probabilities are always greater than or equal to zero.

**Mathematical Expression**: **P(E) ≥ 0**

The second axiom, known as the normalization axiom, states that the probability of the entire sample space 5 is equal to 1. In other words the sum of the probabilities of all possible outcomes is always equal to 1. This axion ensures that the total probability space covers all possible outcomes

**Mathematical Expression: P(S) =1**

The third axiom known as the additivity axiom deals with the probability of the union of two events. It states that the probability of the union of two mutually exclusive events *A* and *B* is equal to the sum of their individual probabilities, *P*(*A*)+*P*(*B*). Mutually exclusive events are events that cannot occur simultaneously.

**Mathematical Expression** (P(A ∪ B) = P(A) + P(B) for mutually exclusive events)

- You have a deck of cards, and you draw two cards without replacement. Calculate the probability of drawing two kings.

**Answer**

The probability of drawing two kings can be calculated as a compound event involving drawing a king on the first draw and a king on the second draw.

- Probability of drawing a king on the first draw:
- There are 4 kings in a deck of 52 cards.
- So, the probability of drawing a king on the first draw is 4/52.

- Probability of drawing a king on the second draw (without replacement):
- After drawing one king, there are now 3 kings left in a deck of 51 cards.
- So, the probability of drawing a king on the second draw is 3/51.

Probability plays a crucial role in our everyday lives because it helps us make sense of and navigate the inherent uncertainty that surrounds us. Here are some key reasons why probability is important in everyday life, along with examples:

**Weather Forecasting:**Probability is central to weather forecasting. Meteorologists use complex models to calculate the likelihood of various weather conditions occurring. For example, when they predict a 60% chance of rain, they are using probability to convey the uncertainty surrounding the weather. This information helps individuals plan their outdoor activities and make decisions such as whether to carry an umbrella.**Games of Chance:**Many recreational activities and games involve an element of chance, from card games like poker to casino games like roulette. Probability theory helps players make informed decisions, such as when to bet, fold, or draw in poker. Casinos also rely on probability to set the odds in their favor while offering players a chance to win.**Decision-Making:**In decision-making, especially when there are multiple options with varying outcomes, probability helps individuals and businesses assess the risks and rewards associated with each choice. For instance, a company may use probability analysis to decide whether to invest in a new product, considering the probability of success and potential profits.**Medicine and Healthcare:**In medicine, probability is used in various ways, such as assessing the effectiveness of treatments, predicting disease outcomes, and calculating the risk of side effects. Doctors often discuss the probability of certain health outcomes with patients when making treatment decisions.**Insurance:**Insurance companies rely heavily on probability to calculate premiums and assess risk. They use statistical models to estimate the likelihood of various events occurring, such as car accidents or property damage, and set insurance rates accordingly.**Stock Market and Investments:**Investors use probability to make informed decisions about buying and selling stocks, bonds, and other financial assets. Probability models help estimate future market movements and assess the risk associated with different investment strategies.

**1.** In a large population, 20% of people have a certain genetic marker. If two people from this population are randomly selected, what is the probability that both of them have the genetic marker?

**Answer**

To find the probability that both of them have the genetic marker, you can use the probability of both events occurring:

P(Both have the marker) = P(First person has the marker) * P(Second person has the marker)

P(Both have the marker) = (0.20) * (0.20) = 0.04.

So, the probability that both of them have the genetic marker is 0.04 or 4%.

**2:** You are given a standard 6-sided die. If you roll it repeatedly until you get a 6, what is the expected number of rolls required?

**Answer**

This is an example of a geometric distribution, and the expected value (mean) of a geometric distribution is given by 1/p, where p is the probability of success (rolling a 6).

p = 1/6 (since there is a 1/6 chance of rolling a 6 on each roll)

Expected number of rolls = 1 / (1/6) = 6 rolls.

So, the expected number of rolls required to get a 6 is 6.

**3.** Consider a manufacturing process that produces light bulbs. The probability of a bulb being defective is 0.05. If you randomly select 10 bulbs from a batch, what is the probability that at least one of them is defective?

**Answer**

To find the probability that at least one of the 10 bulbs is defective, it's easier to find the probability that none of them are defective and then subtract that from 1 (using the complement rule).

P(None are defective) = (0.95)^10 (since the probability of a bulb not being defective is 1 - 0.05 = 0.95)

P(At least one is defective) = 1 - P(None are defective) = 1 - (0.95)^10 ≈ 0.4013.

So, the probability that at least one of the 10 bulbs is defective is approximately 0.4013 or 40.13%.

Probability is a fundamental concept that enables us to quantify uncertainty and make informed decisions in various aspects of life. It serves as a mathematical tool to assess risks, optimize strategies, and predict outcomes. Through the understanding of basic probability concepts, such as random experiments, events, and sample spaces, we can tackle complex scenarios effectively. Probability axioms provide a solid foundation for calculating probabilities, whether in simple or compound events. Moreover, probability notation, including set operations, allows us to express and compute probabilities for a wide range of situations. Real-world applications demonstrate the pervasive influence of probability in fields like weather forecasting, gaming, finance, healthcare, and more.

**Probability Fundamentals**: Probability is the measure of how likely an event is to occur and is expressed as the ratio of favorable outcomes to the total possible outcomes.**Decision-Making Power**: Probability helps in assessing risk, making rational choices, predicting outcomes, and optimizing strategies in uncertain situations.**Random Experiments and Events**: Random experiments have uncertain outcomes, and events represent specific outcomes or combinations.**Probability Notation**: Probability can be expressed using set notation (union, intersection, complement) and symbols (P(A)) to represent events and their probabilities.**Probability Axioms**: Probability follows axioms, including non-negativity and the addition rule for mutually exclusive events.**Real-Life Applications**: Probability plays a crucial role in weather forecasting, gaming, decision-making, healthcare, insurance, and investment.**Practice and Problem Solving**: Exercises and practice questions help reinforce understanding and application of probability concepts.

1. 500 students are taking one or more courses out of chemistry, physics and mathematics. Registration records indicate course enrolment as follows: chemistry (329), physics (186), mathematics (295), chemistry and physics (83), chemistry and mathematics (217), and physics and mathematics (63), How many students are taking all 3 subjects?

- 37
- 43
- 47
- 53

**Answer**

(d)

**Detailed Explanation**

n(C) = 329

n(P) = 186

n(M) = 295

n(C **∩** P) = 83

n(C **∩** M) = 217

n(P **∩** M) = 63

n(P U C U M)=500

We want to find n(C **∩** P **∩** M) (students taking all three subjects).

Using the principle of inclusion-exclusion:

n(C U P U M) = n(C) + n(P) + n(M) - n(C **∩** P + C **∩** M + P **∩** M) + n(C **∩** P **∩** M)

Now, plug in the values:

500 = 329 + 186 + 295 - (83 + 217 + 63) + n(C **∩** P **∩** M)

500 = 810 - 363 + n(C **∩** P **∩** M)

Now, subtract 810 and 363:

500 - n(C **∩** P **∩** M) = 447

n(C **∩** P **∩** M) = 500-447

n(C **∩** P **∩** M) = 53

So, there are 53 students taking all three subjects (chemistry, physics, and mathematics).

2. A 2-digit number must be chosen randomly from all the 2-digit integers between 1 and 100. What is the probability that the chosen integer is not divisible by seven?

a. 12/90

b. 78/90

c. 77/90

d. 13/90

**Answer**

**c) 77/90**

**Detailed Explanation**

There are 90 two-digit numbers in total, 13 of which are divisible by 7: 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, and 98. As a result, the probability that the chosen number is not divisible by 7 is 1 - 13/90 = 77/90. As a result, (C) is the correct answer.

**3.** In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random has a sibling is ______.

a. 0.66

b. 0.066

c. 0.55

d. 0.055

**Answer**

Answer (a)

**Detailed Answer**

Let the number of families with a single child = x The number of families with two children = x Therefore, the total number of children is the society = x + 2x = 3x The number of children picked at random having a sibling will be: ns = 2x The required probability will be: P(s) = 2x/3x P(s) = 0.667

**4.** A fair dice is rolled twice. The probability that an odd number will follow an even number is ________.

a. ½

b. ⅙

c. ⅓

d. ¼

Answer (d)

**Answer:**

Case 1: Even Number followed by Odd Number

- There are three even numbers on a standard die: 2, 4, and 6.
- There are three odd numbers on a standard die: 1, 3, and 5.
- So, for this case, we have 3 possibilities for the first roll (even number) and 3 possibilities for the second roll (odd number).
- There are three odd numbers on a standard die: 1, 3, and 5.
- So, for this case, we have 3 possibilities for the first roll (odd number) and 2 possibilities for the second roll (odd number without repetition).
- There are three even numbers on a standard die: 2, 4, and 6.
- So, for this case, we have 3 possibilities for the first roll (even number) and 2 possibilities for the second roll (even number without repetition).

**5.** A bag contains 4 red balls, 3 green balls, and 5 blue balls. If you reach into the bag without looking and randomly select two balls (without replacement), what is the probability that the first ball is red, and the second ball is green?

a.4/12

b.3/12

c.1/11

d.3/11

**Answer**

Step 1: Probability of selecting a red ball first

- There are 4 red balls out of a total of 12 balls.
- So, the probability of selecting a red ball first is 4/12, which simplifies to 1/3.
- After selecting a red ball, there are now 11 balls left in the bag.
- There are 3 green balls among these 11.
- So, the probability of selecting a green ball second is 3/11.

Step 2: Probability of selecting a green ball second

- After selecting a red ball, there are now 11 balls left in the bag.
- There are 3 green balls among these 11.
- So, the probability of selecting a green ball second is 3/11.

To find the overall probability of both events happening sequentially, we multiply the probabilities from each step:

Probability = (Probability of red first) * (Probability of green second) = (1/3) * (3/11) = 1/11

So, the correct answer is 1/11, which corresponds to option 4.

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