Events in Probability

In Probability, an event can be defined as any outcome or set of outcomes from a random experiment. In other words, an event in probability is a subset of the respective sample space.

From the above images:

1. If you roll a die, the event could be "getting an even number."
2. If you toss two coins simultaneously, the event could be getting "getting at least 1 head".

This concept of events is fundamental to understanding probability theory

Sample Space

A Sample Space is the set of all possible outcomes of an experiment or a random phenomenon. The Sample Space is denoted by the symbol "S" and represents all the possible outcomes that can occur.

Example: When flipping a coin, the sample space is {heads, tails}, because those are the only two possible outcomes. Similarly, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}, because those are the only possible outcomes.

Types of Events in Probability

Main types of events in probability include:

➣ Impossible Event and Sure Event: An event that cannot happen under any circumstances is called an impossible event. Its probability is 0, whereas a sure event (also called a certain event) is an event that is guaranteed to happen, and its probability is always 1.

Examples of Impossible Events:

• Rolling a 7 on a standard 6-sided die.
• Having 30th February in a year.

Examples of Sure Events:

• Rolling a number less than 7 on a die.
• The sun rising in the east.

➣ Independent Event and Dependent Event: Independent events are those in which the probability of an event remains the same, regardless of previous outcomes. Whereas, dependent events are those in which the probability of an event changes based on previous outcomes

Examples of Dependent Events:

Example 1: Drawing two cards from a deck without replacement.

If you draw one card and do not replace it, the total number of cards in the deck changes. The probability of drawing a specific card on the second draw is affected by the outcome of the first draw, hence they are dependent events.

Example 2: Picking a marble from a bag, not replacing it, and then picking another marble.

• If the first marble is not replaced, the total number of marbles changes, which influences the probability of picking the second marble. Hence, the events are dependent.

Examples of Independent Events:

Example 1: Flipping a coin twice.

The outcome of the first flip (heads or tails) does not affect the outcome of the second flip. The probability of each flip remains 1/2, so the events are independent.

Example 2: Rolling a die and flipping a coin.

The result of rolling the die (e.g., getting a 4) has no impact on the result of flipping the coin (heads or tails). Both events are independent.

➣ Simple and Compound Events: When an event consists of only one point of the sample space, this event is called a simple event, and events with two or more points of the sample space are called compound events.

Examples of Simple Events:

Example 1: Tossing a Coin

Sample space: {Heads, Tails}.
Getting "Heads" is a simple event.

Example 2: Rolling a Die

Sample space: {1, 2, 3, 4, 5, 6}.
Getting a 4 is a simple event.

Example of Compound Event:

Scenario: Rolling a fair six-sided die

Let the sample space S be all possible outcomes when rolling the die:
S = {1, 2, 3, 4, 5, 6}

Event A: Rolling an even number (i.e., 2, 4, or 6). So, we can represent it as a set:
A = {2, 4, 6}

Event B: Rolling a number greater than 3 (i.e., 4, 5, or 6). So, we can represent it as a set:
B = {4, 5, 6}

Compound Event (Union of A and B): {2,4,5,6}

➣ Mutually Exclusive Events:  Mutually exclusive events have no outcomes in common. In other words, if one event happens, the other cannot happen.

Example:

Let's consider a scenario of flipping a fair coin.

Event A: The coin lands on heads .A = {Heads}
Event B: The coin lands on tails. B = {Tails}

The intersection of mutually exclusive events is the empty set: A∩B = ∅.

➣ Exhaustive Events:  The collection of those events is exhaustive, covering all the possible outcomes.

Example:

Consider a random experiment where a coin is tossed twice. The sample space (S) is:
S = {HH, HT, TH, TT}

Define the following events:

A: Getting at least one head A = {HH, HT, TH}
B: Getting two tails B = {TT}

The union of A and B gives: A ∪ B = {HH, HT, TH, TT} = S
This means the events A and B together cover all possible outcomes in the sample space S. Hence, A and B are exhaustive events.

➣ Equally Likely Events: Equally likely events have the same probability of occurring. In a random experiment, all outcomes are equally likely if none is favored over the others.

Example:

In the case of rolling a fair six-sided die, there are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. Since all outcomes are equally likely, the probability of rolling any specific number is 1/6.

How to Find the Probability of an Event

• Step 1: Find the total sample space of the experiment.
• Step 2: Find the number of favorable outcomes of the experiment.
• Step 3: Use the formula to calculate the probability as,

Probability = (Favorable Outcome)/(Total Outcome)

Sample Problems on Events in Probability

Problem 1: Let's say a coin is tossed once. State whether the following statement is True or False. 

“If we define an event X, which means getting both heads and tails. This event will be a simple event.”

Solution:

When a coin is tossed, there can be only two outcomes, Heads or Tails. 
S = {H, T} 

Getting both Heads and Tails is not possible, thus event X is an empty set. 
Thus, it is an impossible, not a sure event. So, this statement is False. 

Problem 2: Consider the experiment of tossing a fair coin 3 times. Event A is defined as getting all tails. What kind of event is this? 

Solution: 

Sample space for the coin toss will be, 

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

For the event A, 
A = {TTT}

This event is only mapped to one element of sample space. Thus, it is a simple event. 

Problem 3: A die is thrown in the game of Ludo, and E₁ denotes the event of getting even numbers and E₂ represents the event of getting a number more than 3. Find the Set for the following events,

1. E₁ or E₂
2. E₁ and E₂

Solution:

The sample space for the die will be,
S = {1, 2, 3, 4, 5, 6}

• E₁ (only even numbers) = {2, 4, 6}
• E₂ (number more than 3) = {4, 5, 6}

1. E₁ or E₂ = {2, 4, 5, 6}
2. E₁ and E₂ = {4, 6}

Problem 4: A die is thrown and the set for the sample space obtained is, S = {1, 2, 3, 4, 5, 6} E₁ is defined as the event of obtaining a number less than 5 and E₂ is defined as the event of obtaining a number more than 2.
Find the set for the following,

1. E₁ but not E₂
2. E₂ but not E₁

Solution:

Sample space will be, 
S = {1, 2, 3, 4, 5, 6}

• E₁ (a number less than 5) = {1, 2, 3, 4}
• E₂ (a number more than 2) = {3, 4, 5, 6}

1. E₁ but not E₂ = {1, 2}
2. E₂ but not E₁ = {5, 6}

Problem 5: The sample Space of an experiment is given as,

S = {10, 11, 12, 13, 14, 15, 16, 17} and the event, E is defined as all the even numbers. What will be the complementary event for E?

Solution:

S = {10, 11, 12, 13, 14, 15, 16, 17}
E (All even numbers) = {10, 12, 14, 16}
E' (complementary of E) = {11, 13, 15, 17}

Problem 6: Name the types of events obtained from the experiments given below.

1. A coin is tossed for the 5th time,, and in the event of getting a tail when the first four times, the result was ahead.
2. S (sample space) = {1, 2, 3, 4, 5} and E = {4}
3. S = {1, 2, 3, 4, 5} and E = {2, 4}
4. S = {1, 2, 3, 4, 5}, E₁ = {1, 2} and E₂ = {3, 4}

Solution:

1. No matter how many times the coin is tossed, every time the probability of getting a tail will be 0.5 irrespective of the previous outcomes, therefore the event will be an independent event.

2. E = {4} is a Simple event.

3. E = {2, 4} is a compound event.

4. E₁ and E₂ are Mutually exclusive events.

Problem 7: Write the sample space for tossing three coins at once, also answer the event of 2 exactly 2 heads at a time.

Solution:

Tossing Three Coins the sample space is,
S = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

Hence, the Sample Space Comprises 8 Possible Outcomes
Event (E) for the occurrence of exactly two heads,

E = {(H, H, T), (H, T, H), (T, H, H)}

Problem 8: A die is rolled. Let's define two events: Event A is getting the number 2, and Event B is getting an even number. Are these events mutually exclusive? 

Solution: 

Sample space for die roll will be, S = {1, 2, 3, 4, 5, 6} 

For the event A, 
A = {2}

For the event B, 
B = {2, 4, 6}

For two events to be mutually exclusive, their intersection must be an empty set 

A ∩ B = {2} ∩ {2, 4, 6}
A ∩ B  = {2}

Since it is not an empty set, these events are not mutually exclusive.

Problem 9: A die is rolled, and three events A, B, and C are defined below:

• A: Getting a number greater than 3 
• B: Getting a number that is a multiple of 3. 
• C: Getting an odd number

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

Solution:

Sample space for die roll will be, 
S = {1, 2, 3, 4, 5, 6} 

For the event A, 
A = {4, 5, 6}

For the event B, 

B = {3, 6}

For the event C, 
C = {1, 3, 5}

A ∩ B = {4, 5, 6} ∩ {3, 6}
A ∩ B = {6}

A ∩ B ∩ C = {4, 5, 6} ∩ {3, 6} ∩ {1, 3, 5}
A ∩ B ∩ C = ∅ (Empty Set) 

A ∪ B = {4, 5, 6} ∪ {3, 6}
A ∪ B = {3, 4, 5, 6}

Unsolved Practice Questions:

Question 1: Two coins are tossed simultaneously.
Let: A = event of getting at least one head, B = event of getting both tails.

Find:

1. Sample space S
2. Events A and B
3. Are A and B mutually exclusive?

Question 2: A die is thrown twice. Let E be the event that the sum of the numbers obtained is greater than 8, and F be the event that both numbers are even. List all outcomes in E, F, and E ∩ F.

Question 3: Three coins are tossed together.
Let: A = event of getting exactly two heads, B = event of getting at least one tail.

Find the sets A, B, and A ∩ B, and state whether A and B are independent.

Question 4: A bag contains 5 red balls and 3 blue balls. One ball is drawn at random, its color is noted, and then it is replaced. The experiment is repeated twice.
Let: E1 = event of getting a red ball on the first draw, E2 = event of getting a blue ball on the second draw.

Determine whether E1 and E2 are independent events.

Question 5: A die is rolled once.
Let: A = event of getting an odd number, B = event of getting a number greater than 3, C = event of getting a prime number.

Find:

1. A ∩ B
2. A ∩ C
3. A ∩ B ∩ C
4. Are A, B, and C mutually exclusive?

Related Articles:

• Probability Distribution
• Chance and Probability
• Union of Sets
• Dependent and Independent Events
• Types of Events in Probability