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Probability

Birthday cake

Birthday cake (GMVozd, iStockphoto)

Birthday cake

Birthday cake (GMVozd, iStockphoto)

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Subjects
Let's Talk Science
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8.97

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Learn about the branch of mathematics that helps us predict what might happen known as probability.

Introduction

Mathematics, in a very basic sense, describes relationships between numbers and other measurable quantities. It is called the “language of science.” A strong understanding of mathematics is essential in order to develop a deep understanding of science.

There are many different branches in mathematics, each with many unique and fascinating aspects. In this backgrounder you will explore ways in which we use mathematics to help us to predict what might happen. This is known as This is known as probability. We hope this gets you to think of mathematics in a whole new light!

Probability

Take a coin and flip it. What side would you expect to land facing up? If the coin is fair (hasn’t been rigged to land on a specific side), you would probably say that there is an equal chance of it being heads or tails. We can phrase this statement in terms of probabilities. The probability of the coin landing head side up is one in two (1/2). It can only be one of two possible choices. The probability of it landing tail side up is also one in two (1/2). Note that the sum of the probabilities of all possible outcomes is 1 (= 1/2 +1/2 or 0.5 + 0.5). The outcome of adding to 1 is a fundamental property of probability. 

A question you might ask is “how do you know that these probabilities are correct?” One way to do this is to flip the coin lots of times, record the result each time, and then infer the probabilities from the results. Suppose we flipped the coin 10 times, and obtained the results H,T,H,H,T,T,T,H,T,T (where T stands for “tails” and H for heads). We got four heads and six tails. From this, we might conclude that the probability of getting a head is 4/10 = 0.4 and the probability of getting a tail is 6/10 = 0.6. These values are close to the expected probability of 0.5, but don’t exactly agree. Why is this? It’s because you will only get the expected probabilities if you flip the coin an extremely large number of times. The more times you test, the closer to the expected probability you will get. You can try this with your class!

It is interesting to consider questions that involve combining probabilities. This means we use combinatorics and probability together. For example, suppose we flipped the coin twice, and asked what the probability is of getting tails up two times in a row. For this, we take the probability of the first coin being a tail (1/2) and multiply it by the probability of the second coin being a tail (1/2), leading to a final result of 1/2 x 1/2 = 1/4. 

We can see this by examining all possible outcomes of flipping the coin twice.

Possible combinations from flipping two coins
Possible combinations from flipping two coins (Images from the Royal Canadian Mint via Wikimedia Commons and Wikimedia Commons).

From this we can see that, of the 4 possible outcomes, only one leads to two successive tails, so the probability is 1/4.

Question 1:

A farmer breeds a white cow and a black cow. The calves (baby cows) have an equal chance of being either all white, all black, black with white faces or white with black faces (assume these are the only four possibilities for offspring). The answers are at the end of this Backgrounder.

     a) What is the probability that a calf will be all black?

     b) What is the probability that three calves in a row will be all white?


A famous example of probability is known as the Birthday Problem. What are the chances, in your class, that two students have the same birthday? The answer depends on how large the class is. The chance that this is true is likely a lot higher than you think.

To figure this out, it is actually easier to calculate the probability of two students not having the same birthday. Since the sum of the probability of two students having the same birthday and the probability of two students not having the same birthday is 1, calculating one of these probabilities easily allows us to infer the other. We would just subtract the probability from 1, to get the other probability. In the following example we’ll assume that there are 365 days in a year (so not a leap year).

Let us start with a class of two students. The first person can have any birthday, while the probability that the second person does not have the same birthday is 364/365, because their birthday could be any day except the first person’s birthday. The probability of these two people not having the same birthday is 365/365 x 364/365 = 0.9973, or 99.73%. Thus, the chances of these two students having the same birthday is 1 - 0.9973 = 0.0027, or 0.27% - so, pretty unlikely.

How about 3 students? As before, the first person can have any birthday, and the probability that the second person does not have the same birthday as the first person is 364/365. The probability that the third person does not have a birthday on either of dates of the first two students is 363/365. Thus, the probability that, of these three students, none of them share a common birthday is 365/365 x 364/365 x 363/365 = 0.9918. Thus, the probability that, of these three students, at least two of them do share a common birthday is 1 - 0.9918 = 0.0082, or 0.82%.

You may see the pattern developing – for 4 students, the probability of none of them having the same birthday is 365/365 x 364/365 x 363/365 x 362/365, leading to a probability of at least two of them having the same birthday of about 0.0164, or 1.64%. You might ask “How large must a class be in order for the probability to be 50% of finding two people with the same birthday?” Having seen how this goes, you might want to try calculating it yourself – the answer is that you need only a class of about 23 students!

Question 2:

Given the number of students in your class, what is the probability that two people in your class have the same birthday?

Do any two people in your class have the same birthday?

Try testing this out in your school with various classes to see what the probabilities are. The results will probably amaze most of your classmates, and likely even your teachers! Probabilities may require thinking through a lot of possibilities, but chances are, the results may surprise you.

My Career

Anna Szuto

Genetic counsellor, Division of Clinical and Metabolic Genetics

The Hospital for Sick Children

The Hospital for Sick Children is a specialized hospital for children and their families. Doctors there perform complex surgeries and treat children with cancer. In the Clinical and Metabolic Genetics division, we evaluate and treat children who have, or may be at risk of having, a genetic condition such as Down syndrome or Cystic fibrosis. 

Genetic conditions are caused by mutations, which are variations in a person’s DNA. Mutations can cause the cells in our body to not work properly. Different mutations can cause many different genetic conditions. Each condition has certain symptoms and certain ways of being passed down to the next generation (inheritance). Each has a certain prognosis, which is what will likely happen to a patient who has the condition.

As a genetic counsellor, I use probabilities when I answer patient’s questions about genetic conditions. This may be about the possibility that a genetic condition might explain their symptoms. Or it could be about the probability that they will have a child with a genetic disorder. Or even the probability that they have inherited a condition that is present in a family member. To do this, I collect and analyze my patients’ personal medical histories. I also take a detailed account of the health of their family members, which is known as a family history. I then provide information about the different genetic tests that can be used to establish if my patients have a disease-causing mutation. We talk about disease management as well as about the risks and benefits of genetic testing.

I use the information I’ve collected about the family history and genetic test results to determine the probabilities that my patients might carry mutations. I might also determine the probabilities that their children or other family members could also have inherited the mutation.

My favourite aspect of being a genetic counsellor is meeting new families and helping them to understand the impacts of having a genetic condition on their families and even on their culture.

Anna Szuto
Anna Szuto (© Anna Szuto. Used with permission).

 

Spotlight on Innovation

Probability and Wind Energy

For thousands of years, energy from the wind has been used by people to propel ships, grind grain and pump water. More recently, it has been used to generate electricity using wind turbines. Since the speed and direction of the wind varies from place to place and from day to day, wind speed data is vital to understanding how much energy a wind turbine could produce at a particular site. Wind engineers use tools such as anemometers, which are wind speed measuring devices. They also use data logging systems to collect data about the wind speed at a site before they install a large wind farm. A wind farm is a site with multiple wind turbines.

wind speed probability graph shows the percentage of time the wind blows at a given wind speed. The wind speed is on the x-axis and the probability that the wind will blow at that speed (given in percentages) is on the y-axis.

As with other probabilities, the sum of all of the probabilities equals 1, or 100%. When reading the graph, it is important to note that the taller the bar, the more likely it is that the wind will blow at the speed of that bar.

Understanding the frequency of wind speeds is important when choosing a site for a wind turbine. Notice the wind patterns at location 1 (blue) and location 2 (orange) on the graph above. The two locations have the same average wind speed (5.8 m/s), but very different distributions of frequency. Even though the locations have the same average wind speed, a wind turbine at location 1 is likely to generate more electricity over time because the probability is greater than the wind will blow at higher wind speeds, which enables the turbine to generate more electricity.

To find out more about wind speed patterns in your area, check out the Canadian Wind Energy Atlas.

Wind speed probability graph
Wind speed probability graph (©2020 Let’s Talk Science).

Answers

Question 1: A farmer breeds a white cow and a black bull. The calves (baby cows) have an equal chance of being all white, all black, black with white faces or white with black faces.

a) What is the probability that a calf will be all black?

There is a one in four chance (1/4) that a calf will be all black.

b) What is the probability that three calves in a row will be all white?

There is a one in four chance (1/4) that a calf would be all white. To have three in a row that are all white you would multiple the chances of each being white

1/4 x 1/4 x 1/4 = 1/64 or 0.016 (1.6%) chance

 

 

References

Math is Fun. (n.d.). Probability.

Siegmund, D. O. (n.d.). Probability theory. Encyclopaedia Britannica.

Wolfram MathWorld. (n.d.). Birthday problem. WolframAlpha.

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