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 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 = (Number of Favorable Outcome)/(Total Number of Outcome)
➢Practice: Solved Examples
