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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
There are 8 references cited in this article that you can see at the bottom of the page.
This article has been viewed 204,866 times.
You’ve probably had to calculate probability before, but what exactly is probability, and how is it calculated? Probability is the chance that something will happen, such as winning the lottery or rolling a 6 of the dice. You can easily calculate the probability by using the probability formula (the number of desired outcomes divided by the total number of outcomes). In this article, we will guide you step by step on how to use the probability formula and provide some examples of how to calculate probability through the formula.
Steps
Find the probability of a random event
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Choose an event with mutually exclusive outcomes. Probability can only be calculated when the event will fall into one of two cases: either it happens, or it doesn’t happen. That event and its opposite do not occur at the same time. Rolling the dice for a 5 or a horse winning a race are examples of mutually exclusive events. Either you get a 5 or you don’t; That horse either wins or doesn’t. [1] X Research Source
Example: You cannot calculate the probability of an event: “Both 5 and 6 appear on a roll of the dice.”
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Identify all possible events and outcomes. Let’s say you are calculating the probability of rolling a 3 on a 6-sided dice. In this case, “rolling 3” is the event, and the 6 sided dice can produce one of 6 numbers, so the total number of outcomes will be 6. Thus, we know that the case This has 6 possible events and one for which we are calculating the probability. Here are two other examples to help you navigate:
- Example 1 : What is the probability of picking a day that falls on the weekend when we choose a random day of the week? “Pick a date that falls on a weekend” is the event, and the resulting number is the sum of the days in a week: 7.
- Example 2 : In a jar there are 4 blue marbles, 5 red marbles and 11 white marbles. If we take a random marble out of the jar, what is the probability of getting a red marble? “Pick 1 red marble” is the event, and the resulting number is the total number of marbles in the jar: 20.
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Divide the number of events by the number of possible outcomes. The answer will be the probability of an event occurring. In the case of rolling 3 of the dice, the number of events is 1 (each dice has only one face of 3), and the resulting number is 6. You can also express this relationship as 1 ÷ 6, 1/6, 0.166, or 16.6%. Here’s how to find the probabilities of the above examples: [2] X Research Source
- Example 1 : What is the probability of picking a day that falls on the weekend when we choose a random day of the week? The number of events here will be 2 (since each week has 2 weekends), and the number of outcomes will be 7. The probability will be 2 ÷ 7 = 2/7. You can also express it as 0.285 or 28.5%.
- Example 2 : In a jar there are 4 blue marbles, 5 red marbles and 11 white marbles. If we take a random marble out of the jar, what is the probability of getting a red marble? The number of events in this problem is 5 (because there are 5 red marbles), and the resulting number is 20. The probability will be 5 ÷ 20 = 1/4. You can also express it as 0.25 or 25%.
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Add up the sum of all possible events to make sure it equals 1. The probability of all the possible events added together would have to be 1 or 100%. If you don’t get 100%, chances are you’ve miscalculated because you missed a possible event. Double-check the problem to make sure you didn’t miss any facts. [3] X Research Sources
- For example, the probability of rolling 3 on a 6-sided dice is 1/6, and the probability of rolling all other sides of the dice is also 1/6. Therefore, 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6, i.e. = 100%.
Note: For example, if you forget the number 4 of the dice, the total number of possible outcomes will be only 5/6 or 83%, which means there is a problem.
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Express the probability of an improbable outcome as 0. This means that the event is not likely to happen. Although there is usually no problem that calculates the probability of 0, it is not impossible. [4] X Research Sources
- For example, if you calculate the probability that Easter falls on a Monday in 2020, the answer would be 0 because Easter always falls on a Sunday.
Calculate the probability of many random events
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Calculate each probability separately to compute the independent events. Once you know what those probabilities are, you calculate each event separately. Let’s say you want to know what is the probability of rolling the number 5 twice in a row on a six-sided dice. Know that the probability of rolling a 5 once is 1/6 and the probability of rolling a 5 again is also 1/6. The result of the first does not affect the second. [5] X Research Sources
Note: The probabilities of multiples rolling by the number 5 are called independent events, since the first roll does not affect the outcome of the second roll.
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Consider the effect of previous events when calculating the probabilities of dependent events. If an event has occurred that changes the probability of the second event occurring, then you are calculating the probabilities of the dependent events. For example, if you choose 2 cards from a 52 card deck, when you pick the first card it affects the number of cards available when you pick the second card. To calculate the probability for the second event, you need to subtract 1 from the number of possible outcomes . [6] X Research Source
- Example 1 : Consider the event: 2 cards are drawn at random from the deck. What is the probability of drawing both flip cards? The probability of drawing the first flip card is 13/52, or 1/4. (There are 13 flip cards in each deck.)
- Now then, the probability of drawing a second flip card will be 12/51 because 1 flip card has already been drawn. That means your first action has influenced the second result. If you draw a 3-card and don’t put it back, there will be 1 less flip and 1 card in the deck (51 instead of 52).
- Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If 3 marbles are taken out of the jar at random, what is the probability that the first one is red, the second is blue, and the third is white?
- The probability of getting the first red marble is 5/20, or 1/4. The probability of getting the second blue marble is 4/19 because we have less than 1 marble, but the blue one is not less. The probability of drawing the third white marble is 11/18 because we have drawn 2 marbles.
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Multiply the probabilities of each individual event together. Regardless of whether you calculate the probabilities of independent or dependent events with 2, 3 or 10 outcomes, you can calculate the total probability by multiplying the probabilities of each event together. This calculation will give the probability of multiple events occurring one after the other . So, what is the probability of rolling the number 5 of the six-sided dice twice in a row? , the probability of two independent events is 1/6. So we have 1/6 x 1/6 = 1/36. You can also express it as 0.027 or 2.7%. [7] X Research Sources
- Example 1 : Two cards are drawn at random from a deck. What is the probability that both of those cards are flip cards? The probability of the first event occurring is 13/52. The probability of the second event occurring is 13/52 x 12/51 = 12/204 = 1/17. You can also write this answer 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 out of the jar at random, what is the probability that the first one is red, the second is blue, and the third is white? The probability of the first event occurring is 5/20, the probability of the second event is 4/19, and the probability of the third event is 11/18. The total probability would be 5/20 x 4/19 x 11/18 = 44/1368 = 0.032. You can express it as 3.2%.
Convert odds ratio to probability
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Set the ratio of the positive result as the numerator. Back to the marbles example. Suppose you want to find the probability of getting 1 white marble (out of 11 white marbles) out of a jar of marbles (there are 20 in all). The probability of this happening is the ratio of the probability that it will happen to the probability that it will not happen. Because there are 11 white marbles and 9 non-white marbles, you would write the ratio as 11:9.
- The number 11 represents the probability of getting 1 white marble, the number 9 represents the probability of getting another colored marble.
- Thus, the odds ratio here means the probability of you getting a white marble.
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Add the numbers together to convert probability to probability ratio. It’s pretty easy to do. First, we scale the odds into two separate events: the probability of getting a white marble (11) and the probability of getting another colored marble (9). Add these two numbers to get the total result. Write this result as a probability, with the total number of results just calculated as the denominator.
- The event of getting a white marble is 11, and the event of getting another colored marble is 9. Thus, the total number of outcomes is 11 + 9= 20.
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Finding the probability ratio is similar to calculating the probability of a single event. You’ve worked out a total of 20, and 11 of them are the chances of getting a white marble. Thus, the probability of getting a white marble can be calculated similarly to the probability of a single event. Divide 11 (number of positive outcomes) by 20 (total number of events) to find the probability.
- So, in this example, the probability of getting a white marble would be 11/20. Do the division, we have: 11 ÷ 20 = 0.55 or 55%.
Advice
- You may need to know that in betting on horse racing or sports, odds are often expressed as “adverse odds”, meaning the odds of an event happening are written first, and the odds an event that does not occur is written later. It may seem confusing, but you should know this if you are going to bet on a sporting event.
- The most common ways of expressing probability are written as fractions, decimals, and percentages, or as a scale from 1 to 10.
- Mathematicians often use the term “relative probability” to refer to the probability that an event will occur. The word “relative” is used here because no outcome is 100% guaranteed to happen. For example, if we toss a coin 100 times, the probability of tossing heads and heads will not be exactly 50-50. Relative probability demonstrates this. [8] X Research Sources
- The probability of an event is never negative. If the calculation result is a negative number, you need to check the calculation again.
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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
There are 8 references cited in this article that you can see at the bottom of the page.
This article has been viewed 204,866 times.
You’ve probably had to calculate probability before, but what exactly is probability, and how is it calculated? Probability is the chance that something will happen, such as winning the lottery or rolling a 6 of the dice. You can easily calculate the probability by using the probability formula (the number of desired outcomes divided by the total number of outcomes). In this article, we will guide you step by step on how to use the probability formula and provide some examples of how to calculate probability through the formula.
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