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Probability is a fundamental concept in the field of mathematics and statistics that plays a crucial role in making predictions and assessing uncertainty in various real-world scenarios. It provides a framework to determine the likelihood of an event occurring, enabling us to make informed decisions and analyze risks. Whether you are working on a math problem, conducting a scientific experiment, or making business decisions, understanding how to calculate probability is essential. In this guide, we will explore the foundational principles of probability, learn about different probability models, and discover techniques to estimate and calculate probabilities efficiently. So, if you are ready to dive into the fascinating world of probability, let’s get started!
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
Example: You cannot calculate the probability of an event: “Both 5 and 6 appear on a roll of the dice.”
- 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.
- 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%.
- 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.
- 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
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.
- 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.
- 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
- 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.
- 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.
- 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.
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.
In conclusion, calculating probability is a fundamental concept in the field of mathematics and statistics. It allows us to quantitatively assess the likelihood of events occurring and make informed decisions based on this information. By understanding the basic principles and formulas of probability, such as the probability of an event, the probability of two independent events, and the probability of complementary events, we can apply this knowledge to a wide range of real-life scenarios. Additionally, the concept of conditional probability enables us to determine the likelihood of an event occurring given that another event has already happened. This knowledge is essential in fields such as finance, business, medicine, and engineering, where risk and uncertainty play a significant role. By mastering the techniques and calculations involved in probability, we can analyze data, predict outcomes, and make more informed decisions in various aspects of our lives.
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