You've heard the terms, theoretical probability and experimental probability. but what do they mean? Are they in anyway related? This is what we are going to discover in this lesson.
If you've completed the lessons on independent and dependent probability. then you've already found the theoretical probability for numerous problems.Theoretical Probability
Theoretical probability is the probability that is calculated using math formulas. This is the probability based on math theory.Experimental Probability
Experimental probability is calculated when the actual situation or problem is performed as an experiment. In this case, you would perform the experiment, and use the actual results to determine the probability.
In order to accurately perform an experiment, you must:
Let's take a look at an example where we first calculate the theoretical probability, and then perform the experiment to determine the experimental probability. It will be interesting to compare the theoretical probability and the experimental probability. Do you think the two calculations will be close?
This problem is from Example 1 in the independent events lesson. We calculated the theoretical probability to be 1/12 or 8.3%. Take a look:Like This Page?
Probability theory is an essential subject of mathematics. We study about probabilities in this subject.
The probability may be defined as chances of an event to be happen. It is the numerical measure of the extent of likeliness of occurrence of some event. In other words, to find p robability of an event is to quantify the chances of its happening or not happening.
The probability lies between 0 and 1. The event whose probability is 0 is an impossible event ; while an event with probability 1 is said to be a sure event. Probability can be expressed in the form of fractions and decimals.
There are mainly two types of probabilities:
ii)Experimental Probability As the names themselves suggest that the theoretical probability is based upon theory or formula. On the other hand, the experimental probability is calculated by the actual experiments. Let us go ahead and learn these two types of probabilities and methods of evaluating them in detail.
The theoretical probability is the measure of theoretically happening of an event. This is something which is expected to be happened. Theoretical probability is calculated in a theoretical way. It is considered as the standard. It is not necessary that this probability actually occurs every time. Theoretical probability may differ from the actual occurrence of the event. This probability is estimated using formula. This can be done by dividing the possible number of favorable outcomes by the total outcomes.
Theoretical probability denotes the possible number of ways by which an event can occur. This probability is a method of determining what could happen in a situation, on the basis of given information. It is mere calculation.
By theoretical probability, we may have the idea of something which is likely to occur in a particular situation, but the actual result cannot be predicted by this.
It is calculated by the following formula:
There is another way of finding probability of some event. It is to perform an experiment. The e xperimental probability is the probability of some event which occurs during an actual experiment. Experimental probability can be determined by conducting the experiment. It is not just estimated mathematically.
We have discussed that the t heoretical probability is something that is expected to be occur; while experimental probability is the measure what actually occurs on an experiment. Experimental probability is also calculated by the similar formula described above. But the number of favorable outcomes are found by the real conduction of experiment. I n experimental probability, t he whole event should be performed.
The main difference between theoretical probability and experimental probability is that t heoretical probability is a measure of what should happen; on the other hand, experimental probability is the measure what had happened in actual practice.
The theoretical probability of obtained any number on a roll of die is $\frac<1><6>$. This should ideally happen. But in an actual practice, maybe the die gets that number 2-3 times in successive trials. This may be different than the theoretical probability.
In order to find theoretical probability, one should follow the steps mentioned below:
Step 1: Firstly observe the event closely.
For Example: Let the event be flip of a coin. It should be kept in mind that a coin has two faces - one is called head and another is known as tails.
Step 2: Focus on what to be determined. Let us suppose that in this example, the chances of appearing a tail is asked.
Step 3: Calculated the total number of possible outcomes. Here, the total outcomes would be 2 - head and tail.
Step 4: Now determine theoretically the number of favorable outcomes. In this case, on a coin, favourable outcome is 1 i.e. tail itself.
Step 5: Substitute these values in the formula:
In this example:
The theoretical probability of getting heads and tails are 50% each. In this case, the experimental probability of heads is slightly higher than theoretical one and that of tails is slightly less that theoretical probability.
Example2: Two dice are rolled 50 times and their outcomes are noted as follows :
Sum of the outcomes of both dice are
3, 4, 6, 5, 5, 8. 9, 10, 7, 5, 12, 9, 6, 5, 7, 8, 7, 4, 8, 8, 11, 6, 8. 6, 8, 4, 4, 5, 7, 8, 9, 7, 8, 7, 8, 11, 6, 7, 5, 4, 3, 6, 7, 7, 8, 9, 7, 8, 6, 7
1) Calculate the experimental probability of getting a sum of 8.
2) Find the theoretical probability of getting a sum of 8.
3) Compare both the probabilities.
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Definition Page. Contains definitions arranged alphabetically.
So far we have been plotting and analyzing distributions that come from actual data that we have collected. In the notes, the homework and the quizzes, we have real scores from real samples. These distributions are called empirical distributions. There is another set of distributions called theoretical distributions. These are hypothetical distributions that only appear under certain unusual circumstances that may or may not actually exist. This chapter, on the standard normal curve, and standard scores is based on the information you can get from these theoretical distributions.
The normal distribution
A symmetrical, continuous, and asymptomatic bell shaped distribution of scores
Scores bunch up around the mean
The mean, median, and mode are all the same number
The normal curve
A graph of the normal distribution
The curve never actually touches the x-axis
When graphed, it generally shows 6-8 standard deviations (3-4 one either side of the mean)
If we divide the distribution up into standard deviation units, a known proportion of scores lies within each portion of the curve.
Tables exist so that we can find the proportion of scores above and below any part of the curve, expressed in standard deviation units. Scores expressed in standard deviation units, as we will see shortly, are referred to as Z-scores.
Area under the normal curve
Looking at the figure above, we can see that 34.13% of the scores lie between the mean and 1 standard deviation above the mean. An equal proportion of scores (34.13%) lie between the mean and 1 standard deviation below the mean. We can also see that for a normally distributed variable, approximately two-thirds of the scores lie within one standard deviation of the mean (34.13% + 34.13% = 68.26%).
13.59% of the scores lie between one and two standard deviations above the mean, and between one and two standard deviations below the mean. We can also see that for a normally distributed variable, approximately 95% of the scores lie within two standard deviations of the mean (13.59% + 34.13% + 34.13% + 13.59% = 95.44%).
Finally, we can see that almost all of the scores are within three standard deviations of the mean. (2.14% + 13.59% + 34.13% + 34.13% + 13.59% + 2.14% = 99.72%) We can also find the percentage of scores within three standard deviation units of the mean by subtracting .13% + .13% from 100% (100.00% - (.13% + .13%) = 99.74%). (The difference in these totals 99.72, and 99.74 is due to rounding)
Standard normal distribution (aka, unit normal distribution)
Has a mean of 0
Has a standard deviation of 1
Scores from this distribution are called z scores
Standard scores (z scores)
Changing raw scores to z scores allows for comparisons across measurement tools that are not already equivalent
It is only possible to change raw scores to z scores if the raw scores come from a relatively normally distributed set of scores
Standard score formula
A z of –1.03 is paired with a proportion of .3485 between the score and the mean. Because we are below the mean, we subtract this number from 0.5000 (which is the mean) to get a proportion of 0.1515. a score of 40 is at the 15.15th percentile.
Normal distribution – a theoretical mathematical distribution that specifies the relative frequency of a set of scores in a population.
Standard normal distribution – A normal distribution with a mean of 0 and a standard deviation of 1.
Outcome – each possible occurrence in a probability distribution
Event – the occurrence of a specific set of outcomes in a probability distribution
Probability of occurrence of an event – the number of outcomes comprising an event, divided by the total number of possible outcomes
Discrete outcomes – Outcomes in a distribution that have a countable set of outcomes
Mutually exclusive outcomes – outcomes that cannot occur at the same time
Theoretical probability distribution – a probability distribution found from the use of a theoretical probability model
Empirical probability distribution – a probability distribution found by counting actual occurrences of an event
Standard score – A score obtained by using the transformation z = X - / S
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