Classical probability with Examples
This article will be about the classical probability, how to calculate it with the formula and step by step solved examples.
What is the classical probability
The classical probability predicts a result based on every possible outcome on an aleatory experiment. The classical probability works of a way where the probability is distributed equally in ever possible outcome that compose the sample space, this condition could change if instead of only having individual outcomes we have sets of outcomes, because if we have a set of outcomes, obviously some of this sets will have more chances of happening, but this does not mean that the probability is not distributed equally.
To understand better the general definition of classical probability we are going to take the next example: there is a group of people which are listed by numbers between 1 and 10, and one of them are going to get a price, the decision will be make by picking a random number between 1 and 10 and the person with that number is going to be the winner.
In this case every contestant have the same probability of being the winners of the price that is 1/10 or 10%, but if between this 10 persons there is group of 3 friends (this would be a set of outcomes), then the probability that one of this group of friends is the winner would no longer be 10%, instead the probability for this outcome set would increase to 30%, but again, this does not mean that the probability is distributed unequally between the 10 contestants, is just that there is an outcome set.
Laplace Rule
The formula used in classical probability is also known as the “Laplace rule”, this formula consist divides all the favorable outcomes of an event between the total amount of outcomes. When we have done this we will get a number between 0 and 1, if the result is not between this range then it is possible that we have make a mistake in the process. To convert this value in percentage we have to multiply the number obtained by 100%.
- being A a possible event
- P(A) = n° of favorable outcomesn° total of outcomes
if we get the probability of every outcome, it is possible to confirm that the result is the correct answer, what we have to do is to sum every probability and the result of that sum will have to be 1 (or 100% if is in percentages), If the result is different than 1 there is chance that we made a mistake (results near 1 like 0.999 is also acceptable).
Finding the classical probability
To find the classical probability we are going to use the example of rolling a dice. First we have to find every possible outcome, and we are going to call this a “sample space”, in the case of rolling a dice we already know that we have 6 different outcomes, one for each face of the dice, so we can define the sample space like this: {1,2,3,4,5,6}
Now to find the classical probability of one of this events we can use the formula presented before, in this case we are going to find the probability that when rolling a dice the result is 2.
- Probability for the side 2
- P(2) = 1 / 6 *100%
- P(2) = 0.1666 * 100%
- P(2) = 16.66%
in this example every side has the same probability of 16.66% and if we sum every probability 6 times (that is the total of events) the result will be approximately 100, that means that the answer is correct.
As we explain before, you can also find the probability of a set of outcomes, for example: What is more likely when you roll a dice, to get a prime number or a number greater than 4.
before we develop the this example we have to define the events, for the first one the prime numbers between 1 and 6 are 2, 3 and 4, and the numbers greater than 4 are 5 and 6.
- side = prime number
- P(prm) = 3 / 6 *100%
- P(prm) = 0.5 * 100%
- P(2) = 50%
- side > 4
- P(>4) = 2 / 6 *100%
- P(>4) = 0.3333* 100%
- P(2) = 33.33%
Now that the odds are found we can say that is more likely to get a prime number than a number greater than 4.
Examples of finding the classical probability
Example 1: between 7 people are dealed 5 cards each, the objective of the game is that who obtains the higher combination of card will be the winner ¿What is the probability that each person have to win in the first round?
- w = win
- P(w) = 1 / 7 * 100%
- P(w) = 0.1429* 100%
- P(w) = 14.29%
Example 2: A man thought about a number between 1 and 15, if this man asks his friend to guess the number he has thought, ¿What are the chances for the friend to guess this number in the first try?
- gn = guess the number
- P(gn) = 1 / 15 * 100%
- P(gn) = 0.067 * 100%
- P(gn) = 6.7%
Example 3: A person has the opportunity to earn $100, $200, $500, $800 and $1 000 dollars for spinning a roulette where these quantities are set, the problem is that this person wants to buy himself a new cellphone and for that he needs at least $400, so ¿What are the chances for him to win enough amount of money so he can buy the phone he wants?
First of all we have to define the sample space: {100,200,500,800,1 000}
second we have to define the quantities of favorable outcomes to the event, these are the quantity greater than $400: 500, 800 and 1 000 (a total of 3)
- np = new phone
- P(np) = 3 / 5 * 100%
- P(np) = 0.6* 100%
- P(np) = 60%
after calculating the classical probability we can tell that if he spins the roulette he is more likely to get enough money for his new cellphone.
Example 4: in a group of 100 people 5 prices will be riffle, if between this 100 persons there is a group of 7 friends ¿what are the chances that at least one of these 7 friends wins some of the prices?
First we have to understand that every person has a the same chances to win one of the prices so there are 5 favorable outcomes for every person, so if we sum the 5 favorable outcomes of the 7 persons that belong to the group of friends, we would have a total of 35 favorable outcomes and a total of 100 outcomes.
- Tfo = total of favorable outcomes
- First we sum the favorable outcomes of each friend
- tfo = 5+5+5+5+5+5+5
- tfo = 35
- and then we calculate the probabilty(p7p = probability of the 7 persons)
- P(p7p) = 35 / 100
- P(p7p) = 0.35 * 100%
- P(p7p) = 35%