1.5 Probability of what?

A probability takes a value in the sliding scale from 0 to 100%. Throughout the book we will study how to compute probabilities in many situations. But don’t just focus on computation. Always remember to properly interpret probabilities. This section covers a few ideas to keep in mind when interpreting probabilities17. (We’ll see later how to compute probabilities like those in the examples.)

Example 1.13 In each of the following parts, which of the two probabilities, a or b, is larger, or are they equal? You should answer conceptually without attempting any calculations.

  1. Flip a coin which is known to be fair 10 times.

    1. The probability that the results are, in order, HHHHHHHHHH.
    2. The probability that the results are, in order, HHTHTTTHHT.

  2. Flip a coin which is known to be fair 10 times.

    1. The probability that all 10 flips land on H.
    2. The probability that exactly 5 flips land on H.

Solution. to Example 1.13

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  1. Many people would say the probability in (b) is larger, but the probabilities in (a) and (b) are equal (and both equal to \(\left(\frac{1}{2}\right)^{10} =\frac{1}{1024}\)). The sequence in (b) seems to look “more random”. However, the probability of seeing that particular sequence — H then H then T then H then T… — is the same as seeing the sequence H then H then H then H then H… If the coin is fair and the flips are independent, all possible sequences of flips are equally likely. Think of it this way: choose any flip, say the third. Then that flip is equally likely to be H (as in the third flip for (a)) or T (as in the third flip for (b)). No matter which flip it is, or the results of the other flips, any flip is equally likely to be H or T.

    (Of course, the above response assumes that the coin is fair. If the coin is known to be fair then the sequences in (a) and (b) are equally likely. However, if we actually observed the sequence in (a) we might suspect that the coin is actually not fair. There is an important difference between assumption and observation.)

  2. The probability in (b) is larger. Contrast this to the previous part. There is only one sequence which results in 10 heads, HHHHHHHHHH. However, there are many sequences which result in exactly 5 heads — 252 out of 1024 possibilities in fact — of which HHTHTTTHHT is just one possibility.

Pay close attention to the differences in the two parts in Example 1.13. The first part involves probabilities of the particular outcome sequence. The second part involves more general “events” that the particular outcome sequence might satisfy. The following provides another example of this “particular” versus “general” dichotomy.

Example 1.14 In each of the following parts, which of the two probabilities, a or b, is larger, or are they equal? You should answer conceptually without attempting any calculations.

  1. In the Powerball lottery there are roughly18 300 million possible winning number combinations, all equally likely.

    1. The probability you win the next Powerball lottery if you purchase a single ticket, 4-8-15-16-42, plus the Powerball number, 23.
    2. The probability you win the next Powerball lottery if you purchase a single ticket, 1-2-3-4-5, plus the Powerball number, 6.

  2. Continuing with the Powerball

    1. The probability that the numbers in the winning number are in a row.
    2. The probability that the numbers in the winning number are not in a row.

Solution. to Example 1.14

Show/hide solution
  1. Many people would say the probability in (a) is larger, since the sequence in (a) looks “more random”, but the probabilities in (a) and (b) are equal. Since the outcomes are equally likely, the probability that any single sequence is the winning number is (roughly) 1/300,000,000. (If you don’t believe this, ask yourself: Why would the Powerball conduct its drawing in such a way that some numbers are more likely to be winners than others? And if some numbers were more likely than others, why wouldn’t people know about this?)
  2. The probability in (b) is larger. Contrast this to the previous part. There are only a handful of winning numbers for which the numbers are in a row: 1 through 6, 2 through 7, 3 through 8, etc. However, almost all of the 300 million possibilities do not have numbers in a row.

When interpreting probabilities, be careful not to confuse “the particular” with “the general”.

“The particular:” A very specific event, surprising or not, often has low probability.

  • For a fair coin, observing the particular sequence HHTHTTTHHT in 10 flips is just as likely as observing HHHHHHHHHH.
  • The probability that the winning powerball number is 4-8-15-16-42-(23) is exactly the same as the probability that the winning powerball number is 1-2-3-4-5-(6).
  • The probability that you get a text from your best friend at 7:43pm on Oct 12, 2021 inviting you to dinner after you’ve just ordered pizza from your favorite pizza place is probably pretty small. None of these items — getting a text, having a friend invite you to dinner, ordering pizza from your favorite pizza place — is unusual, but the chances of them all combining in this way at this particular time are fairly small.

“The general:” While a very specific event often has low probability, if there are many like events their combined probability can be high.

  • There are many possible sequences of 10 coin flips which result in 5 heads.
  • For almost all of the possible Poweball combinations the numbers are not in order.
  • The probability that some time in the next month or so a friend invites you for dinner is probably fairly high.

Example 1.15 Consider the following two probabilities.

  1. The probability that you win the next Powerball lottery if you purchase a single ticket.
  2. The probability that someone wins the next Powerball lottery. (FYI: especially when the jackpot is large, there are hundreds of millions of tickets sold.)

Which of these two probabilities is larger, or are they about that same? You should answer conceptually without attempting any calculations.

Solution. to Example 1.15

Show/hide solution

The probability in (2) is much larger. (This is an understatement.)

  • The probability that a specific powerball ticket is the winning number is about 1 in 300 million. So if you buy a single ticket, it is extremely unlikely that you will win.
  • However, if hundreds of millions of powerball tickets are sold, the probability that someone somewhere wins is pretty high.

We elaborate on these ideas below.

The probability that you win the next Powerball lottery if you purchase a single ticket is about 1 in 300 million. Let’s put this number in perspective. There are about 260 million adults (over age 18) in the U.S.19 Suppose that the name of every adult in the U.S. is written on a 3x5 index card. These 260 million cards stacked would stretch about 62 miles high; that’s commonly referenced as the distance from the earth to where space begins. The stack would also weigh about 400 tons, about as much 4 blue whales. Suppose we shuffle the cards (easier said than done) and select one. The probability that your name is on the selected card is about 1 in 260 million. The chances that your next Powerball ticket is the winning number are a little less likely than this20.

However, if hundreds of millions of Powerball tickets are sold, the probability that someone somewhere wins is pretty high. For example, if 500 million tickets are sold then there is a roughly 80% chance that at least one ticket has the winning number (under certain assumptions).

Even if an event has extremely small probability, given enough repetitions of the random phenomenon, the probability that the event occurs on at least one of the repetitions is often high21.

Consider the headline of this news article from 2010: “Man mauled by bear after lightning strike”. We certainly feel sorry for this poor man, but just how unlikely is such an occurrence? Let’s look a little closer.

The headline seems to imply that the man got struck by lightning and then, while he was trying to reach safety, a bear attacked. But the mauling occurred four years after the lightning strike. Getting mauled by a bear and struck by lightning within one’s lifetime is certainly much more likely than both happening on the same day.

“Getting struck by lightning” is often colloquially used to describe a rare event, but how unlikely is it? One study estimates that about 250,000 people in the world are struck by lightning each year, and the National Weather Service estimates that the probability that you get struck by lightning within your lifetime is 1/15,000. Still not very likely, but maybe not as rare as you might think.

Getting mauled by a bear is much mless likely than being struck by lightning. There are only about 40 bear attacks of humans each year. However, if the headline had been “Man bitten by shark after lightning strike” or “Man attacked by mountain lion after lightning strike” or “Man trampled by moose after lightning strike” it probably would have been equally newsworthy. Thus we should account for all similar animal attacks, not just bear attacks, when assessing the likelihood.

The probability that you get struck by lightning and mauled by a bear today is certainly very small. But the probability that someone somewhere within their lifetime gets both struck by lightning and attacked by an animal is orders of magnitude higher. In general, even though the probability that something very specific happens to you today is often extremely small, the probability that something similar happens to someone some time is often quite high.

When assessing a probability, always ask “probability of what”? Does the probability represent “the particular” or “the general”? Is it the probability that the event happens in a single occurrence of the random phenomenon, or the probability that the event happens at least once in many occurrences? Keep these questions in mind when assessing numerical probabilities. Remember that something that has a “one in a million chance” of happening to you today will happen to about 7000 people in the world every day.

1.5.1 Exercises

  1. Create your own analogy for how unlikely that a single ticket wins the Powerball lottery. How would you describe a 1 in 300 million chance?

  2. In each of the following, which is greater: (1) or (2)? Or are they equal? Or is there not enough information to decide?

    1. Election interference
      1. The probability that Russian agents successfully interfere with the 2024 U.S. Presidential election through posts on Facebook with the goal of helping the Republican candidate get elected.
      2. The probability that non-U.S. actors attempt to interfere with the 2024 U.S Presidential election.
    2. Roll a six-sided die which is known to be fair 10 times.
      1. The probability that the results are, in order, 1223334444.
      2. The probability that the results are, in order, 4614253226.
    3. Roll a six-sided die which is known to be fair 10 times.
      1. The probability that the results are, in order, 1234561234.
      2. The probability that you roll each of the six faces at least once.

  1. Read this for a humorous take on interpreting probabilities.↩︎

  2. The exact count is 292,201,338. We will see how to compute this number later.↩︎

  3. Source: U.S. Census Bureau.↩︎

  4. The statistician Ron Wasserstein has provided several fanciful perspectives on the likelihood of winning the Powerball lottery.↩︎

  5. For an interesting investigation of this idea check out the Infinite Monkey Theorem Experiment at the site The Pudding.↩︎