Theoretical Probability calculates the likelihood of an event based on known outcomes, assuming equal likelihood. It's fundamental in predicting probabilities without relying on experimental data.
In this article, we will learn about, theoretical probability, and experimental probability, their differences, examples, applications, and others in detail.
What is Probability?
Probability is the measure of the chance that an event will happen; it is a number between 0 and 1. An event is certain to happen if its probability is one, and zero if it is impossible. In flipping a coin, for example, the chance it will land heads is 0.5. This is so because heads are one of the two possible results. Either by seeing and closely examining facts or by understanding every imaginable scenario, probability may be computed.
There are two ways of studying probability that are
What is Theoretical Probability?
An event's chance of happening is measured theoretically using known probable outcomes. It is computed by dividing the number of favourable results by all feasible outcomes. With this approach, any result is assumed to be equally probable. On a fair six-sided die, for instance, the theoretical chance of rolling a 3 is 1/6 since there is only one favourable result—rolling a 3—and six potential outcomes—rolling any number from 1 to 6.
Theoretical Probability
In the image added above the theoretical probability of getting 7 when the wheel is move is, 1/8.
Many disciplines employ theoretical probability to forecast results in place of experimental evidence.
Theoretical Probability Definition
Theoretical probability is the likelihood of an event occurring, calculated by dividing the number of favourable outcomes by the total number of possible outcomes, assuming each outcome is equally likely.
Theoretical Probability for Event A is calculated by formula:
P(E) = (Number of Event in an Experiment) / (Total Number of Trials)
How to Find Theoretical Probability?
Clearly define the experiment or random process you are thinking about. Determine the Sample Space by making a list of every experiment result.
Select Advantageous Results: Determine which particular results satisfy the requirements for the event you are considering.
Count Total Outcomes: Total up all the potential outcomes in the sample space.
Count Positive Results: Total up the positive results.
Compute Probabilities: Amount of positive outcomes divided by total outcomes.
Use Formula: P (Event) = (Number of Favourable Outcomes)/ (Total Number of Outcomes)
Reduce the fraction to its simplest words.
Write Probability: Put the probability as a percentage, decimal or fraction.
Examples on Theoretical Probability
Various examples on Theoretical Probability are added below:
1. Rolling a Fair Six-Sided Die
A standard six-sided die has faces numbered from 1 to 6. The probability of rolling a specific number, such as a 4, is calculated as follows:
Number of favorable outcomes: 1 (since there is only one 4 on the die)
Total number of possible outcomes: 6 (since the die has 6 faces)
Probability of Rolling a 4 = 1/6
2. Drawing a Card from a Standard Deck
A standard deck of cards has 52 cards. To find the probability of drawing an Ace:
Number of favorable outcomes: 4 (since there are 4 Aces in the deck)
Total number of possible outcomes: 52 (total cards in the deck)
Probability of drawing an Ace = 4/52 = 1/13
3. Flipping a Fair Coin
A fair coin has two sides:
Probability of getting heads on a single flip is:
Number of Favorable Outcomes: 1 (since there is only one heads)
Total number of possible outcomes: 2 (heads or tails)
Probability of Getting Heads = 1/2
4. Drawing a Ball from a Bag
Suppose you have a bag with 3 red balls, 2 blue balls, and 5 green balls. The total number of balls is 10. The probability of drawing a blue ball is:
Number of favorable outcomes: 2 (since there are 2 blue balls)
Total number of possible outcomes: 10 (total balls in the bag)
Probability of drawing a blue ball = 2/10 = 1/5
5. Rolling Two Dice
When rolling two six-sided dice, there are 36 possible outcomes (6 faces on the first die multiplied by 6 faces on the second die). To find the probability of rolling a sum of 7:
Combinations that give a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)
Number of favorable outcomes: 6
Total number of possible outcomes: 36
Probability of rolling a sum of 7 = 6/36 = 1/6
6. Choosing a Student Randomly
Imagine a classroom with 12 boys and 8 girls. The probability of randomly selecting a girl from the class is:
Number of favorable outcomes: 8 (since there are 8 girls)
Total number of possible outcomes: 20 (12 boys + 8 girls)
Probability of selecting a girl = 8/20 = 2/5
7. Drawing a Marble from a Jar
If a jar contains 4 red marbles, 5 blue marbles, and 1 yellow marble, the probability of drawing a yellow marble is:
Number of favorable outcomes: 1 (since there is only 1 yellow marble)
Total number of possible outcomes: 10 (4 red + 5 blue + 1 yellow)
Probability of drawing a yellow marble = 1/10
8. Picking a Letter from a Word
Consider the word "PROBABILITY". There are 11 letters in total. The probability of picking the letter 'B' at random is:
Number of favorable outcomes: 2 (since the letter 'B' appears twice)
Total number of possible outcomes: 11 (total letters in the word)
Probability of picking a 'B' = 2/11
Experimental Probability Definition
Experimental probability is the chance of a happening given the real outcomes of a trial or experiment. It is computed by dividing the overall number of trials carried out by the frequency of the occurrence.
Experimental Probability Example
Example: Suppose you roll a six-sided die 100 times, and you observe that the number 4 comes up 18 times.
Solution:
Experimental probability of rolling a 4 is:
Number of Times Event Occurs (rolling a 4): 18
Total Number of Trials (total rolls): 100
Experimental Probability = (Number of Times Eevent Occurs) / (Total Number of Trials)
= 18 / 100 = 0.18
So, the experimental probability of rolling a 4 on a six-sided die in this experiment is 0.18.
Theoretical Probability Vs Experimental Probability
Aspect
| Theoretical Probability
| Experimental Probability
|
|---|
Definition
| Based on the expected likelihood of events occurring
| Based on the actual results from experiments or trials
|
|---|
Calculation Method
| Number of favourable outcomes divided by total possible outcomes
| Number of successful outcomes divided by total trials
|
|---|
Basis
| Assumes all outcomes are equally likely
| Relies on empirical data gathered from experiments
|
|---|
Example
| Probability of rolling a 3 on a fair six-sided die is 1/6
| Probability calculated after rolling a die 100 times
|
|---|
Reliability
| Idealized, assumes perfect conditions | Reflects real-world variability and experimental error
|
|---|
Use Case
| Used in theoretical models, predictions, and simulations
| Used to verify theoretical models and in practical applications
|
|---|
Dependence on Data
| No need for actual data, purely based on possible outcomes
| Requires collection of data from repeated experiments
|
|---|
Variability
| Fixed value for a given scenario | Can vary depending on the experimental conditions
|
|---|
The image added below shows the basic difference between Experimental probability and Theoretical probability.
Comparing Experimental and Theoretical Probability
We can compare experimental probability and theoretical probability by the example added below:
Theoretical Probability Example: Probability of getting 3 on a die.
Probability of rolling a 3 on a fair six-sided die:
Number of favorable outcomes: 1 (only one 3 on the die)
Total number of possible outcomes: 6
Theoretical Probability = 1/6 ≈ 0.167
Experimental Probability Example: Probability when rolling a six-sided die 60 times results in a 3 appearing 12 times.
Number of times the event occurs (rolling a 3): 12
Total number of trials (total rolls): 60
Experimental probability = 12/60 = 0.20
In this example, the theoretical probability is 0.167, while the experimental probability is 0.20. Differences between the two can arise due to the randomness and finite number of trials in the experiment.
Conclusion
In conclusion, theoretical probability is based on expected outcomes under ideal conditions, while experimental probability is derived from actual experimental results. Both methods are essential in different contexts: theoretical probability for predictions and models, and experimental probability for real-world applications and validation.
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