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| Terms Related to Probability | Measuring Empirical Probability | |
Chapter 15 Probability (Concepts)
Welcome to the fascinating study of Probability, a branch of mathematics dedicated to quantifying uncertainty and measuring the likelihood of various occurrences. In everyday life, we constantly encounter situations where the outcome is not predetermined – the chance of rain, the result of a game, the outcome of a medical test. Probability provides us with a formal framework to analyze these 'chances'. This chapter aims to formalize the study of probability, transitioning from intuitive ideas towards a more structured understanding, primarily focusing on an experimental or empirical approach based on observations and data.
A crucial distinction we explore is between two perspectives on probability:
- Experimental Probability: This is derived from the results of actually performing an experiment multiple times or analyzing recorded data from past observations. It reflects what has happened in a series of trials.
- Theoretical Probability: (More deeply explored in Class 10) This relies on logical reasoning about the nature of the experiment itself, assuming all possible outcomes are equally likely (like a fair coin toss). It predicts what should happen in an ideal scenario.
To apply this, we first need to understand the related terminology. An experiment refers to an action or process with uncertain results (e.g., tossing a coin, rolling a die). Each repetition of the experiment is called a trial. An event is a specific outcome or a collection of some possible outcomes of the experiment that we are interested in (e.g., getting a 'Head', rolling an even number). The set of all possible outcomes of an experiment constitutes the sample space (though formal set theory notation might be limited at this stage). We will engage in simple experiments or analyze given datasets – like records of numerous coin tosses or dice rolls – to calculate the experimental probabilities of various events based on the observed frequencies.
Through these calculations and analyses, we reinforce fundamental properties of probability:
- The probability of any event E must always lie between 0 and 1, inclusive. Mathematically, this is expressed as $\mathbf{0 \le P(E) \le 1}$.
- If $P(E) = 0$, the event E is considered an impossible event (it cannot happen based on the observed trials or theoretical setup).
- If $P(E) = 1$, the event E is a certain event or sure event (it is guaranteed to happen based on the observed trials or setup).
- The sum of the probabilities of all possible elementary events (single outcomes) in an experiment is always equal to 1.
Basic Terms Related to Probability
In many situations in life, the result of an action is not certain. For example, when you toss a coin, you cannot predict with absolute certainty whether it will land as 'Heads' or 'Tails'. When you roll a dice, you don't know which number will show up. These are examples of situations involving randomness or chance.
Probability is a branch of mathematics that provides a way to measure and quantify the likelihood or chance of an event occurring. It allows us to express uncertainty using numerical values.
In this chapter, we will focus on Empirical Probability (also known as Experimental Probability or Observed Probability), which is based on the results of actual experiments or observations.
Key Terms in Probability
To understand and calculate probability, it is important to be familiar with the following basic terms:
Experiment:
An experiment is an action or operation that is performed, and which produces one or more possible results or outcomes. The outcome of a particular performance of the experiment cannot be predicted with certainty beforehand.
Examples of experiments: Tossing a coin, rolling a standard six-sided dice, drawing a card from a well-shuffled deck of cards, surveying a group of people.
Trial:
A trial is a single performance or execution of an experiment.
Example: Tossing a coin one time is a trial. Rolling a dice one time is a trial. Drawing one card from a deck is a trial.
Outcome:
An outcome is one of the possible results of a trial or experiment.
Examples:
- In the experiment of tossing a coin, the possible outcomes are Head (H) or Tail (T).
- In the experiment of rolling a standard six-sided dice, the possible outcomes are getting a 1, 2, 3, 4, 5, or 6.
Sample Space:
The sample space of an experiment is the set of all possible outcomes of that experiment. It lists every potential result.
Examples:
- The sample space for tossing a single coin is S = {Head, Tail} or {H, T}.
- The sample space for rolling a standard six-sided dice is S = {1, 2, 3, 4, 5, 6}.
- The sample space for tossing two coins simultaneously is S = {HH, HT, TH, TT}.
Event:
An event is a specific outcome or a collection of one or more outcomes from the sample space. It is a result or a set of results that we are interested in.
Examples:
- In the experiment of tossing a coin, 'getting a Head' is an event. 'Getting a Tail' is another event.
- In the experiment of rolling a dice, 'getting an even number' is an event. The outcomes included in this event are {2, 4, 6}.
- In rolling a dice, 'getting a number less than 3' is an event. The outcomes included are {1, 2}.
- In tossing two coins, 'getting at least one head' is an event. The outcomes are {HH, HT, TH}.
Favourable Outcome(s):
The favourable outcome(s) for a specific event are those outcome(s) from the sample space that meet the criteria or condition of the event we are interested in.
Example: In rolling a dice, for the event "getting an even number", the favourable outcomes are 2, 4, and 6. The number of favourable outcomes is 3.
Empirical Probability (or Experimental Probability):
Empirical probability is determined by conducting an experiment a number of times and observing the outcomes. It is based on actual results obtained from trials.
The empirical probability of an event E is calculated using the formula:
P(E) $= \frac{\text{Number of times the event E happened}}{\text{Total number of trials in the experiment}}$
This is the type of probability we will primarily focus on in Class 9, where we deal with results from conducted experiments or collected data.
Theoretical Probability (or Classical Probability):
Theoretical probability is based on the assumption that all possible outcomes of an experiment are equally likely to occur. It is calculated by analysing the total possible outcomes and the favourable outcomes without actually performing the experiment.
The theoretical probability of an event E is given by:
P(E) $= \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$
This concept is usually explored in more detail in later classes.
Range of Probability:
The probability of any event E, P(E), is always a value between 0 and 1, inclusive.
$0 \le P(E) \le 1$
- If P(E) = 0, the event is considered an impossible event (it cannot happen).
- If P(E) = 1, the event is considered a certain event (it will definitely happen).
Complementary Event:
The complementary event of an event E is the event that E does not occur. It is usually denoted by E' or $\overline{E}$.
The sum of the probability of an event E and the probability of its complementary event E' is always equal to 1.
$P(E) + P(E') = 1$
This implies that $P(E') = 1 - P(E)$.
Example 1. In the experiment of rolling a standard six-sided dice, describe the experiment, a trial, possible outcomes, and the event of getting a number greater than 4.
Answer:
- The experiment is the process or action being performed, which is rolling a standard six-sided dice.
- A trial is a single instance of performing the experiment, i.e., rolling the dice once.
- The possible outcomes are the individual results that can occur when the dice is rolled. For a standard six-sided dice, the possible outcomes are the integers from 1 to 6. So, the possible outcomes are 1, 2, 3, 4, 5, or 6.
- The event of getting a number greater than 4 is a specific set of desired outcomes from the possible outcomes. The numbers greater than 4 on a standard dice are 5 and 6. So, the event is the set of outcomes {5, 6}. The favourable outcomes for this event are 5 and 6.
Measuring Empirical Probability
As discussed in the previous section, Empirical Probability (also known as Experimental Probability or Observed Probability) is a way to determine the likelihood of an event based on the results of actual experiments or observations. It relies on conducting trials of an experiment and recording how many times a specific event occurs.
Formula for Empirical Probability:
To calculate the empirical probability of an event, we use the following formula:
Let $E$ be a specific event we are interested in (e.g., getting a Head when tossing a coin, getting a 4 when rolling a dice). We conduct an experiment a certain number of times (total trials). We count how many times the event $E$ occurs during these trials.
The empirical probability of event $E$, denoted by $P(E)$, is given by:
$P(E) = \frac{\text{Number of trials in which event E happened}}{\text{Total number of trials}}$
This formula gives us a numerical value representing the estimated chance of the event occurring based on the conducted experiment.
Properties of Probability:
Regardless of whether probability is empirical or theoretical, it follows certain fundamental properties:
- Range of Probability: The probability of any event $E$ is always a value between 0 and 1, inclusive. This means it cannot be negative or greater than 1.
- Probability of an Impossible Event: An impossible event is an event that has no chance of occurring. The probability of an impossible event is always 0. For example, the probability of rolling a 7 on a standard six-sided dice is 0.
- Probability of a Sure Event: A sure event (or certain event) is an event that is guaranteed to occur. The probability of a sure event is always 1. For example, the probability of getting a number less than 7 when rolling a standard six-sided dice is 1.
- Sum of Probabilities of All Outcomes: The sum of the probabilities of all possible outcomes of an experiment is always equal to 1. This is because, in any trial, one of the possible outcomes must occur. If the outcomes are mutually exclusive (cannot happen at the same time) and exhaustive (cover all possibilities), the sum of their individual probabilities is 1.
$0 \leq P(E) \leq 1$
Example 1. A coin is tossed 500 times with the following frequencies:
Head: 245 times
Tail: 255 times
Find the empirical probability of getting a Head and the empirical probability of getting a Tail.
Answer:
Given:
Total number of coin tosses (trials) = 500.
Number of Heads = 245.
Number of Tails = 255.
To Find:
Empirical probability of getting a Head.
Empirical probability of getting a Tail.
Solution:
Empirical Probability of getting a Head:
Let $E_{\text{Head}}$ be the event of getting a Head.
The number of times the event $E_{\text{Head}}$ happened is the number of Heads obtained, which is 245.
Using the formula for empirical probability:
$P(E_{\text{Head}}) = P(\text{Head}) = \frac{\text{Number of Heads}}{\text{Total number of tosses}}$
Substitute the given values:
$P(\text{Head}) = \frac{245}{500}$
Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 5:
$P(\text{Head}) = \frac{\cancel{245}^{49}}{\cancel{500}_{100}}$
$P(\text{Head}) = \frac{49}{100} = 0.49$
... (1)
The empirical probability of getting a Head is 0.49.
Empirical Probability of getting a Tail:
Let $E_{\text{Tail}}$ be the event of getting a Tail.
The number of times the event $E_{\text{Tail}}$ happened is the number of Tails obtained, which is 255.
Using the formula for empirical probability:
$P(E_{\text{Tail}}) = P(\text{Tail}) = \frac{\text{Number of Tails}}{\text{Total number of tosses}}$
Substitute the given values:
$P(\text{Tail}) = \frac{255}{500}$
Simplify the fraction by dividing by 5:
$P(\text{Tail}) = \frac{\cancel{255}^{51}}{\cancel{500}_{100}}$
$P(\text{Tail}) = \frac{51}{100} = 0.51$
... (2)
The empirical probability of getting a Tail is 0.51.
As a check, note that the sum of probabilities of all possible outcomes (Head and Tail) is $0.49 + 0.51 = 1$, which aligns with the properties of probability.
Example 2. A dice is rolled 100 times and the outcomes are recorded as follows:
| Outcome (Number on Dice) | Frequency |
|---|---|
| 1 | 15 |
| 2 | 20 |
| 3 | 18 |
| 4 | 16 |
| 5 | 14 |
| 6 | 17 |
Find the empirical probability of:
(i) Getting a 4
(ii) Getting an even number
(iii) Getting a number less than 3
Answer:
Given:
Results of rolling a dice 100 times, provided in the frequency table.
| Outcome (Number on Dice) | Frequency |
|---|---|
| 1 | 15 |
| 2 | 20 |
| 3 | 18 |
| 4 | 16 |
| 5 | 14 |
| 6 | 17 |
To Find:
Empirical probability for events (i), (ii), and (iii).
Solution:
The total number of trials (dice rolls) is the sum of all frequencies:
Total trials $= 15 + 20 + 18 + 16 + 14 + 17$
Total trials $= 100$
... (1)
(i) Empirical Probability of getting a 4:
Let $E_1$ be the event of getting a 4. From the table, the frequency of getting a 4 is 16.
Number of times $E_1$ occurred = 16
Using the empirical probability formula:
$P(E_1) = P(\text{getting a 4}) = \frac{\text{Frequency of 4}}{\text{Total number of rolls}}$
Substitute the values:
$P(\text{getting a 4}) = \frac{16}{100} = 0.16$
... (2)
The empirical probability of getting a 4 is 0.16.
(ii) Empirical Probability of getting an even number:
Let $E_2$ be the event of getting an even number. The even outcomes when rolling a dice are 2, 4, and 6.
The number of times event $E_2$ occurred is the sum of the frequencies of outcomes 2, 4, and 6.
Number of even outcomes $= \text{Frequency of 2} + \ $$ \text{Frequency of 4} + \ $$ \text{Frequency of 6}$
Number of even outcomes $= 20 + 16 + 17 = 53$
... (3)
Using the empirical probability formula:
$P(E_2) = P(\text{getting an even number}) = \frac{\text{Number of even outcomes}}{\text{Total number of rolls}}$
Substitute the values from equations (1) and (3):
$P(\text{getting an even number}) = \frac{53}{100} = 0.53$
... (4)
The empirical probability of getting an even number is 0.53.
(iii) Empirical Probability of getting a number less than 3:
Let $E_3$ be the event of getting a number less than 3. The outcomes less than 3 when rolling a dice are 1 and 2.
The number of times event $E_3$ occurred is the sum of the frequencies of outcomes 1 and 2.
Number of outcomes less than 3 $= \text{Frequency of 1} + \text{Frequency of 2}$
Number of outcomes less than 3 $= 15 + 20 = 35$
... (5)
Using the empirical probability formula:
$P(E_3) = P(\text{getting a number} < 3) = \frac{\text{Number of outcomes} < 3}{\text{Total number of rolls}}$
Substitute the values from equations (1) and (5):
$P(\text{getting a number} < 3) = \frac{35}{100} = 0.35$
... (6)
The empirical probability of getting a number less than 3 is 0.35.