Probability and odds are constantly being misused. Even in respected publications you will see sentences such as: “The odds of a Red Sox win tonight is 60%.” or “His probability is 2 to 1.” While in the lexicon of American English these words seem to have taken on interchanging meaning, when working with statisticians or data scientists, you are going to want to get your vocabulary straight.
Probability:
Probability is a number between 0 and 1, often represented as a fraction or percentage. The probability is determined by dividing the number of positive outcomes by the total number of possible outcomes. So if you have 4 doors and 1 has a prize behind it, your probability of picking the right door is 1/4 or 0.25 or 25%.
Note, do not let the term positive outcome confuse you. It is not a qualifier of good vs bad or happy vs sad. It simply means the result is what you are looking for based on the framing of your question. If I were to state that out of every 6 patients who opt for this new surgery 2 die – 2 would be the “positive outcome” in my equation (2/6 or approx 33%) even though dying is far from a “positive outcome”.
Odds:
Odds, on the other hand, are a ratio. The odds of rolling a 4 on a six sided die is 1:5 (read 1 in 5). The odds ratio works like this: positive outcomes : negative outcomes. So the odds of rolling an even number on a six sided die is 3:3 (or simplified to 1:1).
Now the probability of rolling an even number on a six sided die is 3/6 or 1/2. So keep that in mind, odds of 1:2 is actually a probability of 1/3 not 1/2.
Deck of Cards:
Working with a standard deck of playing cards (52 cards).
Pulling a Red card from the deck
- probability: 26/52 = 1/2
- odds: 26:26 = 1:1
Pulling an Ace from the deck
- probability: 4/52 = 1/13
- odds: 4:48 = 1:12
Pulling a Diamond from the deck
- probability: 13/52 = 1/4
- odds: 13:39 = 1:3