13.2: Probabilities are Numbers

We think in terms of probabilities more than you might suppose. We often do this, for example, when we talk about percentages. When something could just as easily turn out either of two ways, we say that it’s a 50/50 proposition. We could express the same claim by saying that there is a probability of .5 that either of the two outcomes will occur. The meteorologist says that there is a 65% chance of showers later today. We could instead say that the probability of showers is .65. In general, we can translate claims about percentages into corresponding claims about probability by dividing the percentage by 100; this just means adding a decimal point (and perhaps one or more zeros) at the appropriate place. Thus 90% means a probability of .9, and 3% means a probability of .03.

Polls and surveys also report percentages that can be translated into probabilities. Many of the sciences, from psychology to genetics to physics, make heavy use of probability and statistics (which is itself based on the theory of probability). We are all concerned with the likelihood of various possibilities every day. Is rain so likely that we should cancel our hike? Are allergy shots sufficiently likely to help with my allergies that they are worth the time and bother? How likely is this prisoner to commit another crime if they are paroled this year? How likely is Wilbur to go out with Wilma if she asks him for a date?

Probabilities are numbers that represent the likelihood that something will happen. Probabilities are measured on a scale from 0 to 1. If something is certain to happen, it has a 100% chance of occurring, and we say that it has a probability of 1. And if something is certain not to happen, it has a 0% chance of occurring, and we say that it has a probability of 0. The numbers 0 (“no way”) and 1 (“for sure”) nail down the end points of this scale, so it is impossible to have probability values of 13, 2, or 54. Since life is usually uncertain, we are most often dealing with probabilities greater than 0 and less than 1.

Notation

In our first example, the probability of drawing a red jellybean is 2/5. We might write this as: “Probability (Drawing a red jellybean) = 2/5.” But it will save a lot of writing if we introduce two sorts of abbreviations. First, we abbreviate the word ‘probability’ with ‘Pr’. Second, we abbreviate sentences by capital letters. You can use any letters you please (so long as you do not use one letter to abbreviate two different sentences in a problem). But it’s best to pick a letter that helps you remember the original sentence. For example, it would be natural to abbreviate the sentence, ‘I rolled a six’ by ‘S’.

If we abbreviate the sentence, ‘I drew a red jellybean’ as ‘R’, we can write our claim that the probability of drawing a red jellybean is 25 like this:

Pr(R) = ⅖

If we flip a fair coin it is equally likely to come up heads or tails, so we say that the probability of getting heads on the next toss is .5. We write this as: Pr(H) = .5 (or Pr(H) = 1/2). In general, we write:

Pr(S) = n

to mean that the sentence S has a probability of n of being true.

Exercises

Use this notation to express the following claims:

1. The probability of rolling a six is 1/6.
2. The probability of drawing an ace of spaces from a full deck is 1/52.
3. The probability of rolling a five or a six is 2/6 (i.e., 1/3).
4. The probability of drawing either a red jellybean or a green jellybean is 1.
5. The probability of getting both a red and green jellybean on the same draw is 0.

Answer

There isn’t a uniquely correct way to abbreviate the simple sentences in this exercise, but the following ways are natural.

1. Pr(S) = 1/6
2. Pr(As) = 1/52
3. We have to sneak an ‘or’ into our abbreviated sentence: Pr(F or S) = 1/3
4. Pr(R or G) = 1
5. Pr(R and G) = 0