When calculating probability, it is about determining how likely it is that a specific event will occur given a certain number of attempts. Probability equals the chance that one or more events will occur divided by the number of possible outcomes. To find the probability of multiple events, you must break the problem down into separate probabilities and multiply them by one another.
Method 1 of 3: Finding the Probability of a Single Random Event
Step 1. Choose an event whose results are mutually exclusive
The probability can only be calculated when the event whose probability you are calculating occurs or not. The event and its opposite cannot occur at the same time. Examples of mutually exclusive events are getting a 5 on a die and having a certain horse win a race, since one gets a 5 or does not get it and the horse wins or does not win.
For example, it would not be possible to calculate the probability of an event that is formulated in this way: "By rolling the die only once, you will get a 5 and a 6."
Step 2. Define all the events and outcomes that could occur
Suppose you are trying to find the probability of rolling a 3 on a 6-sided die. "Getting a 3" constitutes the event and, since we know that any of the 6 numbers can be obtained on a 6-sided die, the number of results is 6. So, we know that, in this case, there are 6 possible events and 1 outcome whose probability we are interested in calculating. These are 2 more examples that will help you to orient yourself:
- Example 1: What is the probability of choosing a day that corresponds to the weekend when randomly choosing a day of the week? "Choose a day that corresponds to the weekend" is the event and the number of results is the total number of days of the week (7).
- Example 2: a jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is taken from the jar at random, what is the probability that the marble is red? "Pick a red marble" is the event and the number of results is the total number of marbles in the jar (20).
Step 3. Divide the number of events by the number of possible outcomes
In this way, you will obtain the probability that a single event occurs. In the case of getting a 3 on the die, the number of events is 1 (there is only 3 on each die) and the number of outcomes is 6. This relationship can also be expressed as 1 ÷ 6, 1/6, 0, 166 or 16.6%. This is how the probability is found in the remaining examples:
- Example 1: What is the probability of choosing a day that corresponds to the weekend when randomly choosing a day of the week? The number of events is 2 (since there are 2 days of the week that correspond to the weekend) and the number of results is 7. The probability is 2 ÷ 7 = 2/7. It is also possible to express it as 0.285 or 28.5%.
- Example 2: a jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is taken from the jar at random, what is the probability that the marble is red? The number of events is 5 (since there are 5 red marbles) and the number of outcomes is 20. The probability is 5 ÷ 20 = 1/4. It is also possible to express it as 0, 25 or 25%.
Step 4. Add up the probabilities of all the possible events so that you make sure they are equal to 1
The probability of all possible events must add up to 1 or 100%. If the probability of all possible events does not add up to 100%, it is most likely that you made a mistake by omitting one. Recheck your trades to make sure you haven't missed any possible results.
- For example, the probability of rolling a 3 on a 6-sided die is 1/6, although the probability of rolling the other 5 numbers on a die is also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6, which is equal to 100%.
- For example, if you had forgotten the number 4 on the die, adding up all the probabilities would give you only 5/6 or 83%, which indicates a problem.
Step 5. Represent the probability of an impossible outcome with a 0
This just means that there is no probability that an event will occur and it occurs every time you have an event that simply cannot occur. You are unlikely to find a probability of 9, but it is not impossible either.
For example, if you were to calculate the probability that Easter falls on a Monday in 2020, the probability will be 0 because this holiday always falls on a Sunday
Method 2 of 3: Calculate the Probability of Multiple Random Events
Step 1. Take care of each probability separately to calculate the independent events
After determining what these probabilities are, you will need to calculate them separately. Suppose you want to know the probability of rolling a 5 twice in a row on a 6-sided die. You know that the probability of getting a 5 is 1/6 and of getting another 5 on the same die is 1/6. The first result does not interfere with the second.
The probability of getting two 5s is known as independent events, since what you get the first time has no effect on what happens the second time
Step 2. Consider the effect of previous events when calculating the probability of dependent events
In the event that one event modifies the probability of a second event occurring, you will calculate the probability of dependent events. For example, if you choose 2 cards from a deck of 52 cards, the first card you choose will have an effect on the cards that are available when choosing the second. If you want to calculate the probability of the second of two dependent events, it will be necessary to subtract 1 from the number of possible outcomes when calculating the probability of the second event.
Example 1- Two random cards are drawn from a deck of cards. What is the probability that they are both clubs? The probability that the top card is Clubs is 13/52 or 1/4 (there are 13 Clubs cards in each deck).
Now, the probability that the second card is of clubs is 12/51, since a card of clubs will have been removed. This is because what you do the first time has an effect on the second. In case you take a 3 of clubs and don't return it, there will be one less card of clubs and one less card in the deck (51 and not 52)
Example 2: a jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If 3 marbles are taken from the jar at random, what is the probability that the first is red, the second is blue, and the third is white?
The probability that the first marble is red is 5/20 or 1/4. The probability that the second marble is blue is 4/19, since there is one less marble but there is not one less blue marble, and the probability that the third marble is white is 11/18, since they will have been taken already two marbles
Step 3. Multiply the probabilities of each separate event by each other
Whether you are dealing with independent or dependent events and are working with 2, 3, or even 10 outcomes in total, it is possible to calculate the total probability by multiplying the separate probabilities of the events. In this way, you obtain the probability that several events occur one after the other. So for the scenario: what is the probability of rolling two 5s consecutively on a 6-sided die?, the probability of both independent events is 1/6. With this we get 1/6 x 1/6 = 1/36. It is also possible to express it as 0, 027 or 2.7%.
- Example 1- Two random cards are drawn from a deck of cards. What is the probability that they are both clubs? The probability of the first event occurring is 13/52. The probability of the second occurring is 12/51. The probability is 13/52 x 12/51 = 12/204 = 1/17. It is also possible to express it as 0.058 or 5.8%.
- Example 2: a jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If 3 marbles are taken from the jar at random, what is the probability that the first is red, the second is blue, and the third is white? The probability of the first event is 5/20, that of the second is 4/19, and that of the third is 11/18. The probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032. You can also express it as 3.2%.
Method 3 of 3: Convert Odds to Odds
Step 1. Establish the odds as a proportion with the positive result being the numerator
For example, let's go back to the example of the colored marbles. Suppose you want to determine the probability of taking a white marble (of which there are 11) from the total pool of marbles (in which there are 20). The rate of the event occurring is the ratio between the probability that it does occur divided by the probability that it does not occur. There are 11 white marbles and 9 non-white marbles, so you will write the ratio as the 11: 9 ratio.
- The number 11 represents the probability of choosing a white marble and the number 9 represents the probability of choosing a marble of another color.
- So, you are likely to take a white marble.
Step 2. Add the numbers to convert the odds into probabilities
It is quite simple to convert odds into probabilities. First, you must divide the odds into 2 separate events: the odds of taking a white marble (11) and that of taking a marble of another color (9). Add the numbers to find the number of total results. You should write it as a probability so that the total number of outcomes you just calculated is the denominator.
The event "take a white marble" is 11 and the event "take another color" is 9. The total number of results is 11 + 9 or 20
Step 3. Find the odds as if you were calculating the probability of a single event
You have calculated that there are 20 possibilities in total and that, in essence, 11 of these results correspond to taking a white marble. So now you can approach the probability of taking a white marble as if it were any other probability calculation for a single event. Divide 11 (the number of positive results) by 20 (the number of events in total) to get the probability.
So, in our example, the probability of taking a white marble will be 11/20. By dividing it, you will get 11 ÷ 20 = 0, 55 or 55%
- Mathematicians generally use the term "relative probability" to refer to the probabilities that an event will occur. They include the word "relative" because no result is 100% guaranteed. For example, if you flip a coin 100 times, it is probable that you don't get exactly 50 heads and 50 stamps. Relative probability takes this into account.
- The probability of an event must always be a positive number. In case you get a negative number, you should double check your calculations.
- Common ways of writing probabilities include fractions, decimals, percents, or scales from 1 to 10.
- You may need to know that in the sports betting and gaming industry, odds are expressed as "odds against." This means that the odds of an event occurring are written first and the odds that it does not occur are written later. It can be confusing, but it is important that you know this in case you intend to bet on a sporting event.