1.3 Working with probabilities

In the previous section we saw two different interpretations of probability: relative frequency and subjective. Fortunately, the mathematics of probability work the same way regardless of the interpretation. Also, even with subjective probabilities it is helpful to consider what might happen in a simulation.

1.3.1 Consistency requirements

With either the relative frequency or personal probability interpretation there are some basic logical consistency requirements6 which probabilities need to satisfy.

Example 1.1 As of Sept 18, the website fivethirtyeight.com listed the following probabilities for who will win the 2019 World Series.

Houston Astros 25%
Los Angeles Dodgers 21%
New York Yankees 20%
Atlanta Braves 8%
  1. Is the relative frequency or subjective probability interpretation more appropriate here?
  2. According to the site, what must be the probability that the Houston Astros do not win?
  3. According to this site, what must be the probability that one of the above four teams is the World Series champion?
  4. According to this site, what must be the probability that a team other than the above four teams is the World Series champion?

Solution to Example 1.1

  1. It is more appropriate to think of the probabilities themselves in application as subjective. Different websites or models could reasonably assign other probabilities to the teams. However, we can still imagine the probabilities as relative frequencies, if it helps our intuition. If we think of this as a simulation, each repetition results in a World Series champion and in the long run the Astros would be the champion in 25% of repetitions.
  2. Either the Astros win or they don’t; if there’s a 25% chance that the Astros win, there must be a 75% chance that they do not win. If we think of this as a simulation, each repetition results in either the Astros winning or not, so if they win in 25% of repetitions, they must not win in the other 75% to account for 100% of the repetitions.
  3. There is only one World Series champion, so if say the Astros win then no other team can win. Thinking again of a simulation, the repetitions in which the Astros win are distinct from those in which the Dodgers win. So if the Astros win in 25% of repetitions and the Dodgers wins in 21% repetitions, then on a total of 46% of repetitions either the Astros or Dodgers win. Adding the four probabilities, we see that the probability that one of the four teams above wins must be 74%.
  4. Either one of the four teams above wins, or some other team wins. If there is a 74% chance that the winner is one of the four teams above, then there must be a 26% chance that the winner is not one of these four teams.

1.3.2 Odds

The words “probability”, “chance”, “likelihood”, and “odds” are colloquially treated as synonyms. However, in the mathematical language of probability, odds provide a different way of reporting a probability. Rather than reporting probability on a 0% to 100% scale, odds report probabilities in terms of ratios.

Example 1.2 In Example 1.1 the odds that the Astros win the World Series are 3 to 1 against.

  1. What do you think that “3 to 1 against” means?
  2. What are the odds of the Astros not winning?
  3. What are the odds of the Yankees winning?
  4. What are the odds of the Braves winning?

Solution to Example 1.2

  1. The probability that the Astros win is 0.25, so the probability that they do not win is 0.75. These numbers are in a 3 to 1 ratio: the probability of not winning (0.75) is 3 times greater than the probability of winning (0.25). So the odds against the Astros winning the World Series are 3 to 1; “against” because the Astros are less likely to win than to not win.
  2. The probabilities are still in the 3 to 1 ratio, but we can say that the odds are 3 to 1 in favor of the Astros not winning.
  3. The probability that the Yankees win is 0.2 and that they don’t win is 0.8, and \(0.8/0.2 = 4\)). So the odds are 4 to 1 against the Yankees winning (“against” because the Yankees are less likely to win than to not win).
  4. The probability that the Braves win is 0.08 and that they don’t win is 0.92, and \(0.92/0.08 = 11.5\)). So the odds are 11.5 to 1, or 23 to 2, against the Braves winning (“against” because the Braves are less likely to win than to not win).
  • The odds of an event is a ratio involving the probability that the event occurs and the probability that the event does not occur \[ \begin{aligned} \text{odds in favor} & = \frac{\text{probability that the event occurs}}{\text{probability that the event does not occur}} \\ & \\ \text{odds against} & = \frac{\text{probability that the event does not occur}}{\text{probability that the event occurs}}\end{aligned} \]
  • In many situations (e.g., gambling) odds are implicitly reported as odds against.
  • Odds are usually expressed as whole numbers, e.g., 11 to 1, 7 to 2.

Ron and Leslie make the following bet. If Boston wins, Leslie will pay Ron $200; if not, Ron will pay Leslie $100. If both consider this to be a fair bet, what have they agreed that the probability that Boston wins is?

The odds of a fair bet on whether or not an event will occur imply a probability for the event \[\text{probability that event occurs} = \frac{\text{odds in favor of the event}}{1+\text{odds in favor of the event}}\]


  1. In Section 2.4, we will formalize these requirements in the axioms of probability.