Probability – An Experimental Approach: Definition, Examples, Problems

Probability refers to the possibility of a particular event happening. A probability has a number between \(0\) and \(1,\) denoting that if an event is impossible, the likelihood is \(0,\) and if it is certain, the probability is \(1.\) Experimental probability is the probability that is calculated based on the findings of an experiment. Empirical probability is another name for this.

Probability

The probability of an event occurring is defined by probability. There are many situations in which we must determine the outcome of an event in real life. We are certain or uncertain about the outcome. In such conditions, we say that the event has a chance of happening or not happening.

In general, probability has a lot of uses in games and in business to create probability-based predictions.
Formula to find the probability of an event is given by the
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{favourable\;outcomes}}}}{{{\rm{total\;outcomes}}}}.\)

Different Approaches of Probability

There are two approaches of probability. They are,
1. Theoretical Approach
2. Experimental Approach

Theoretical Approach of Probability

A measure of how likely a given outcome (the event) is for a particular experiment, such as tossing a coin, is named the theoretical probability of an event. It’s based on what we could expect to happen if we did not do the experiment. The theoretical approach is also called a classical approach.

The theoretical probability of an event \(E,\) written as \(P\left( E \right),\) is defined as

\(P\left( E \right){\rm{ = }}\frac{{{\rm{number\;of\;outcomes\;favourable\;to\;}}E}}{{{\rm{number\;of\;all\;possible\;outcomes\;of\;the\;experiment}}}}{\rm{.}}\)

Here we assume that the outcomes of the experiment are equally likely.

Experimental Approach of Probability

A probability that is determined by a series of tests is known as experimental probability. Experimental probability, also known as empirical probability, is based on real-world experiments and through recordings of events. A set of actual experiments are conducted to determine the occurrence of any event. Random experiments are those that do not have a specific conclusion.

A random experiment is carried out and repeated many times to determine their probability, with each repetition known as a trial. The experiment is conducted to determine the likelihood of an event occurring or not happening.

Examples: Tossing a coin, rolling a die, etc.

The experimental probability formula is defined mathematically as:

\({\rm{Probability\; = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)

Examples for Experiments in Probability

Tossing a Coin: There are two possible outcomes while tossing a coin. They are head and tail.

The experimental probability of getting head or tail is given mathematically as:

\({\rm{Probability}}\left( {{\rm{head}}} \right){\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}},\) and

\({\rm{Probability}}\left( {{\rm{tail}}} \right){\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}.\)

Throwing Dice: Many games use dice to decide how players move across the board. Dice contains six possible outcomes, and the outcomes of dice are a game of chance that can be determined using probability concepts.

There are six possible outcomes when the dice is thrown are \(1,2,3,4,5,6.\)

While throwing dice, the probability of getting each number is the same, as their probability equals \(\frac{1}{6}.\)

\({\rm{Probability\;of\;getting\;a\;number\;1}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

\({\rm{Probability\;of\;getting\;a\;number\;2}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

\({\rm{Probability\;of\;getting\;a\;number\;3}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

\({\rm{Probability\;of\;getting\;a\;number\;4}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

\({\rm{Probability\;of\;getting\;a\;number\;5}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

\({\rm{Probability\;of\;getting\;a\;number\;6}}\) on dice \({\rm{ = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{6}}}\)

Drawing Cards:: A deck of \(52\) playing cards is divided into \(4\) suits: club, diamond, heart, and spade. Let’s discuss the chances of drawing cards from a pack now.

Below are the symbols on the cards. Hearts and diamonds are red cards, whereas spades and clubs are black.

Every suit of a deck of playing cards consists of \(13\) cards. Each suit consists of the cards ace, \(2, 3, 4, 5, 6, 7, 8, 9, 10,\) king, queen and jack.

The probability of drawing a single card, two or more cards, can be calculated using the same probability formula.

Experimental Versus Theoretical Probability

Solved Examples: Probability – An Experimental Approach

Q.1. A coin is tossed \(1000\) times with the following frequencies: Head: \(460,\) tail: \(540.\) Compute the probability of getting head.
Ans: The total number of trials is \(1000\) because the coin is tossed \(1000\) times.
The number of times heads occur is \(460.\)
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)
\( \Rightarrow {\rm{Probability}}\left( {{\rm{head}}} \right){\rm{ = }}\frac{{{\rm{460}}}}{{{\rm{1000}}}}{\rm{ = }}\frac{{{\rm{23}}}}{{{\rm{50}}}}\)
\( \Rightarrow {\rm{Probability}}\left( {{\rm{head}}} \right){\rm{ = }}\frac{{{\rm{23}}}}{{{\rm{50}}}}\)
Therefore, the probability of getting head is \(\frac{{23}}{{50}}.\)

Q.2. Dice is thrown once. Find the probability of getting a number greater than \(3.\)
Ans: There are six possible outcomes are \(1,2,3,4,5,6.\)
The possible outcomes greater than \(3\) on a die are \(4,5,6.\)
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)
\( \Rightarrow P = \frac{3}{6} = \frac{1}{2}.\)
Hence, the probability of getting a number greater than \(3\) on dice is \(\frac{1}{2}.\)

Q.3. Find the experimental probability in a throw of dice of obtaining a number less than \(5.\)
Ans: There are six possible outcomes in a die are \(1,2,3,4,5,6.\)
The possible outcomes on a die for obtaining a number less than \(5\) are \(1,2,3,4.\)
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)
\( \Rightarrow P = \frac{4}{6} = \frac{2}{3}\)
Hence, the probability of getting an even number on dice is \(\frac{2}{3}.\)

Q.4. Find the experimental probability in a throw of dice of obtaining an even number.
Ans: There are six possible outcomes in a die are \(1,2,3,4,5,6.\)
Even numbers on a die are \(2,4,6.\)
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)
\( \Rightarrow P = \frac{3}{6} = \frac{1}{2}\)
Hence, the probability of getting an even number on dice is \(\frac{1}{2}.\)

Q.5. A coin is tossed \(1000\) times with the following frequencies: Head: \(460,\) tail: \(540.\) Compute the probability of getting tail.
Ans: The total number of trials is \(1000\) because the coin is tossed \(1000\) times.
Let’s label the events of getting a tail, respectively. The number of times tail occurs, that is, \(540.\)
\({\rm{Probability\;of\;an\;event = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)
\( \Rightarrow {\rm{Probability\;}}\left( {{\rm{tail}}} \right){\rm{ = }}\frac{{{\rm{540}}}}{{{\rm{1000}}}}\)
\( \Rightarrow {\rm{Probability\;}}\left( {{\rm{tail}}} \right){\rm{ = }}\frac{{{\rm{27}}}}{{{\rm{50}}}}\)
Therefore, the probability of getting head is \(\frac{{27}}{{50}}.\)

Summary

Experimental probability is a probability that is determined through a series of tests. A random experiment is carried out and repeated many times to determine their probability, with each repetition known as a trial. This article includes the definition of probability, types of probabilities, experimental probability, theoretical probability, examples of experimental probability, experimental versus theoretical probability.

This article, “Probability – an Experimental Approach”, help in understanding these concepts in detail, and it helps solve the problems based on these very easily.

Frequently Asked Questions (FAQs)

Q.1. What are experimental outcomes probabilities?
Ans: A probability that is determined by a series of tests is known as experimental probability. A random experiment is carried out and repeated many times to determine their probability, with each repetition known as a trial. The experiment is conducted to determine the likelihood of an event occurring or not happening.

Q.2. What is an experimental probability example?
Ans: An example of the experimental probability is tossing a coin.

Q.3. What is the set of all possible outcomes of a probability experiment?
Ans: The sample space is the set of all possible outcomes of an experiment. Events are subsets of the sample space that are given a probability of \(0\) to \(1,\) inclusive.

Q.4. What happens to experimental probability as the number of trials increases?
Ans: The experimental probability approaches the theoretical probability as the number of trials increases.

Q.5. Where is experimental probability used?
Ans: Experimental probability is frequently used in social science, behavioural science, economics, and medical research and tests. We can rely on experimental probability in cases where the theoretical probability cannot be determined.

Q.6. Is experimental probability the same as theoretical probability?
Ans: The probability of an event occurring based on experimental results is known as experimental probability. Theoretical probability, on the other hand, is the expected probability of an event occurring.

Q.7. How to find the experimental probability?
Ans: An experiment is repeated a certain number of times, with each repetition usually known as a trial.
The experimental probability formula is defined mathematically as: \({\rm{Probability\; = }}\frac{{{\rm{number\;of\;times\;an\;event\;occurs\;}}}}{{{\rm{total\;outcomes\;of\;trials}}}}\)

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