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Probability

Birthday cake

Birthday cake (GMVozd, iStockphoto)

Birthday cake

Birthday cake (GMVozd, iStockphoto)

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Let's Talk Science
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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
<p>Possible combinations from flipping two coins (Images from the Royal Canadian Mint via <a href="https://en.wikipedia.org/wiki/File:Toonie_-_back.png">Wikimedia Commons</a> and <a href="https://en.wikipedia.org/wiki/File:Toonie_-_front.png">Wikimedia Commons</a>).</p>

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.

    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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