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One of the most important concepts in pure mathematics is probability. The probability theory provides a much better understanding, which leads to possible applications in everyday life. It is a tool used in the sciences and engineering to quantify and reduce uncertainty in the real world. It provides the theoretical foundation for random number generation, statistics, finance and insurance or decision making. The theory of probability has its roots in games of chance and wagers made by gamblers, where it was first recognised that some outcomes were more likely than others without an explanation of why.
Theory of probability:
The theory of probability seeks to explain how we can measure the relative frequency with which an event occurs, given our knowledge about the relevant sample space. For example, in the toss of a fair coin, the sample space is {heads, tails}. If we are looking at ten tosses of a particular coin, we can think of this as creating a finite “experiment” or “measurement” in the sample space of possible outcomes in the long run.
Basic concepts of probability:
A basic probability is a chance that something will happen. Probabilities are not limited to just one event happening or not happening; they can be used to describe anything from the likelihood of whether your team will win, whether your stomach will grow in length during pregnancy, or whether a coin flip lands on heads or tails.
You can think of probability as being like a fraction. One is the numerator of the fraction, and the denominator is 100 per cent. The numerator is what we call our chances or possibilities, and the denominator is how many ways it could happen.
The following fundamental experiments explain the basic theory of probability of an event to carry out many other experiments or measurements.
- Experiment I: In the first experiment, we draw a card randomly from a well-shuffled deck of 52 playing cards.
Let be the event that we draw an ace.
Then these two events can be represented by P( ) and Q( ).
- Experiment II: In the second experiment, we draw a card randomly from a well-shuffled deck of 52 playing cards.
Let be the event that we draw an ace, and let be the event that we draw an odd card.
- Experiment III: We rolled a fair die 20 times in the third experiment. Let be the event that a four is rolled in any one of these rolls.
Then can be represented by P( ).
Understanding theory of probability with fraction concept:
Probability is defined in percentage, which can also be explained as infractions.
You can think of a basic probability as being like a fraction. One is the numerator of the fraction, and the denominator is 100 per cent. The numerator is what we call our chances or possibilities, and the denominator is how many ways it could happen.
Let’s look at how we can use fractions to talk about probabilities. Imagine you have an apple, and two sections of this apple are both green.
It might only have one green section, or it could be divided into two sections. You could take a slice and cut it with scissors in half or use a knife to divide the apple down the middle.
In either case, your chance that this is one of two sections is 50 per cent because there are two ways to do it – cut it in half with a knife or divide it through the middle with scissors.
If you had a second apple with two green sections, then your chances of it being either one or the other would be 50 per cent. (rough theoretical estimate used only for explaining the concept).
It is an example of using fractions to talk about the probabilities of things happening, called the idea of ‘fractional randomness’. It works because the chance is the same, whether it’s one or two sections of an apple.
Now, what if you had an apple that was already divided? If you cut your apple in half, there are only two equal pieces.
The chance of getting one piece instead of the other is not 50 per cent. It is 25 per cent because there are two ways to cut it and two pieces at the end.
Conclusion:
The theory of probability is a branch of mathematics used for determining the likelihood of events. This can calculate the likelihood that a particular event will occur if repeated repeatedly, such as flipping a coin or rolling dice. Alternatively, it can be used to determine the likelihood that an event does not occur, such as throwing dice so that it does not come up with snake eyes (two ones). The probability of an event, which can be calculated using either technique, is the likelihood that the event will occur multiplied by the number of times it could occur.
