Experimental Probability
Experimental probability: introduction, experimental probability: definition, experimental probability formula, solved examples, practice problems, frequently asked questions.
In mathematics, probability refers to the chance of occurrence of a specific event. Probability can be measured on a scale from 0 to 1. The probability is 0 for an impossible event. The probability is 1 if the occurrence of the event is certain.
There are two approaches to study probability: experimental and theoretical.
Suppose you and your friend toss a coin to decide who gets the first turn to ride a new bicycle. You choose “heads” and your friend chooses “tails.”
Can you guess who will win? No! You have $\frac{1}{2}$ a chance of winning and so does your friend. This is theoretical since you are predicting the outcome based on what is expected to happen and not on the basis of outcomes of an experiment.
So, what is the experimental probability? Experimental probability is calculated by repeating an experiment and observing the outcomes. Let’s understand this a little better.
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Experimental probability, or empirical probability, is the probability calculated by performing actual experiments and gathering or recording the necessary information. How would you define an experiment? The math definition of an experiment is “a process or procedure that can be repeated and that has a set of well-defined possible results or outcomes.”
Consider the same example. Suppose you flip the coin 50 times to see whether you get heads or tails, and you record the outcomes. Suppose you get heads 20 times and tails 30 times. Then the probability calculated using these outcomes is experimental probability. Here, t he experimental meaning is connected with such experiments used to determine the probability of an event.
Now that you know the meaning of experimental probability, let’s understand its formula.
Experimental Probability for an Event A can be calculated as follows:
P(E) $= \frac{Number of occurance of the event A}{Total number of trials}$
Let’s understand this with the help of the last example.
A coin is flipped a total of 50 times. Heads appeared 20 times. Now, what is the experimental probability of getting heads?
E xperimental probability of getting heads $= \frac{Number of occurrences}{Total number of trials}$
P (Heads) $= \frac{20}{50} = \frac{2}{5}$
P (Tails) $= \frac{30}{50} = \frac{3}{5}$
Experimental Probability vs. Theoretical Probability
Theoretical probability expresses what is expected. On the other hand, experimental probability explains how frequently an event occurred in an experiment.
If you roll a die, the theoretical probability of getting any particular number, say 3, is $\frac{1}{6}$.
However, if you roll the die 100 times and record how many times 3 appears on top, say 65 times, then the experimental probability of getting 3 is $\frac{65}{100}$.
Theoretical probability for Event A can be calculated as follows:
P(A) $= \frac{Number of outcomes favorable to Event A}{Number of possible outcomes}$
In the example of flipping a coin, the theoretical probability of the occurrence of heads (or tails) on tossing a coin is
P(H) $= \frac{1}{2}$ and P(T) $= \frac{1}{2}$ (since possible outcomes are $2 -$ head or tail)
Experimental Probability: Examples
Let’s take a look at some of the examples of experimental probability .
Example 1: Ben tried to toss a ping-pong ball in a cup using 10 trials, out of which he succeeded 4 times.
P(win) $= \frac{Number of success}{Number of trials}$
$= \frac{4}{10}$
$= \frac{2}{5}$
Example 2: Two students are playing a game of die. They want to know how many times they land on 2 on the dice if the die is rolled 20 times in a row.
The experimental probability of rolling a 2
$= \frac{Number of times 2 appeared}{Number of trials}$
$= \frac{5}{20}$
$= \frac{1}{4}$
1. Probability of an event always lies between 0 and 1.
2. You can also express the probability as a decimal and a percentage.
Experimental probability is a probability that is determined by the results of a series of experiments. Learn more such interesting concepts at SplashLearn .
1. Leo tosses a coin 25 times and observes that the “head” appears 10 times. What is the experimental probability of getting a head?
P(Head) $= \frac{Number of times heads appeared}{Total number of trials}$
$= \frac{10}{25}$
$= \frac{2}{5}$
$= 0.4$
2. The number of cakes a baker makes per day in a week is given as 7, 8, 6, 10, 2, 8, 3. What is the probability that the baker makes less than 6 cakes the next day?
Solution:
Number of cakes baked each day in a week $= 7, 8, 6, 10, 2, 8, 3$
Out of 7 days, there were 2 days (highlighted in bold) on which the baker made less than 6 cookies.
P$(< 6 $cookies$) = \frac{2}{7}$
3. The chart below shows the number of times a number was shown on the face of a tossed die. What was the probability of getting a 3 in this experiment?
Number of times 3 showed $= 7$
Number of tosses $= 30$
P(3) $= \frac{7}{30}$
4. John kicked a ball 20 times. He kicked 16 field goals and missed 4 times . What is the experimental probability that John will kick a field goal during the game?
Solution:
John succeeded in kicking 16 field goals. He attempted to kick a field goal 20 times.
So, the number of trials $= 20$
John’s experimental probability of kicking a field goal $= \frac{Successful outcomes} {Trials attempted} = \frac{16}{20}$
$= \frac{4}{5}$
$= 0.8$ or $80%$
5. James recorded the color of bikes crossing his street. Of the 500 bikes, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were gray. What is the probability that the car crossing his street is white?
Number of white bikes $= 100$
Total number of bikes $= 500$
P(white bike) $= \frac{100}{500} = \frac{1}{5}$
Attend this quiz & Test your knowledge.
In a class, a student is chosen randomly in five trials to participate in 5 different events. Out of chosen students, 3 were girls and 2 were boys. What is the experimental probability of choosing a boy in the next event?
A manufacturer makes 1000 tablets every month. after inspecting 100 tablets, the manufacturer found that 30 tablets were defective. what is the probability that you will buy a defective tablet, the 3 coins are tossed 1000 times simultaneously and we get three tails $= 160$, two tails $= 260$, one tail $= 320$, no tails $= 260$. what is the probability of occurrence of two tails, the table below shows the colors of shirts sold in a clothing store on a particular day and their respective frequencies. use the table to answer the questions that follow. what is the probability of selling a blue shirt.
Jason leaves for work at the same time each day. Over a period of 327 working days, on his way to work, he had to wait for a train at the railway crossing for 68 days. What is the experimental probability that Jason has to wait for a train on his way to work?
What is the importance of experimental probability?
Experimental probability is widely used in research and experiments in various fields, such as medicine, social sciences, investing, and weather forecasting.
Is experimental probability always accurate?
Predictions based on experimental probability are less reliable than those based on theoretical probability.
Can experimental probability change every time the experiment is performed?
Since the experimental probability is based on the actual results of an experiment, it can change when the results of an experiment change.
What is theoretical probability?
The theoretical probability is calculated by finding the ratio of the number of favorable outcomes to the total number of probable outcomes.
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What is experimental probability?
Practice questions, experimental probability – explanation & examples.
Experimental probability is the probability determined based on the results from performing the particular experiment.
In this lesson we will go through:
- The meaning of experimental probability
- How to find experimental probability
The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.
Experimental Probability can be expressed mathematically as:
$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$
Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$. You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice.
Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$.
How do we find experimental probability?
Now that we understand what is meant by experimental probability, let’s go through how it is found.
To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment.
Let’s go through some examples.
Example 1: There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?
Number of coins showing Heads: 12
Total number of coins flipped: 20
$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$
Example 2: The tally chart below shows the number of times a number was shown on the face of a tossed die.
a. What was the probability of a 3 in this experiment?
b. What was the probability of a prime number?
First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events.
a. Number of times 3 showed = 7
Number of tosses = 30
$P(\text{3}) = \frac{7}{30}$
b. Frequency of primes = 6 + 7 + 2 = 15
Number of trials = 30
$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$
Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples.
Example 3: The table shows the attendance schedule of an employee for the month of May.
a. What is the probability that the employee is absent?
b. How many times would we expect the employee to be present in June?
a. The employee was absent three times and the number of days in this experiment was 31. Therefore:
$P(\text{Absent}) = \frac{3}{31}$
b. We expect the employee to be absent
$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June
Example 4: Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey.
a. What is the probability that a car is red?
b. If a new car is bought by someone in town, what color do you think it would be? Explain.
a. Number of red cars = 50
Total number of cars = 500
$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$
b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability.
Now it is time for you to try these examples.
The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.
- What is the probability of selecting a brown jeans?
- What is the probability of selecting a blue or a white jeans?
On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?
Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons.
a. What is the experimental probability of a comedian winning a season?
b. From the next 10 seasons, how many winners do you expect to be dancers?
Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?
Number of brown jeans = 25
Total Number of jeans = 125
$P(\text{brown}) = \frac{25}{125} = \frac{1}{5}$
Number of jeans that are blue or white = 75 + 20 = 95
$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$
Number of beef burgers = 110
Number of burgers (or sandwiches) sold = 200
$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$
a. Number of comedian winners = 3
Number of seasons = 20
$P(\text{comedian}) = \frac{3}{20}$
b. First find the experimental probability that the winner is a dancer.
Number of winners that are dancers = 2
$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$
Therefore we expect
$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.
To find your P(tail) in 10 trials, complete the following with the number of tails you got.
$P(\text{tail}) = \frac{\text{number of tails}}{10}$
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