Odds against/in favor of an event

"Odds against" an event is the ratio of the probability of not happening of that event to the probability of happening of that event.

For example, odds against getting a particular number, say 5, on rolling a number cube is the ratio of the probability of not getting 5 to the probability of getting 5 on rolling that number cube.

"Odds in favor of" an event is the ratio of the probability of happening of that event to the probability of not happening of that event.

For example, odds in favor of getting the number 5 on rolling a number cube is the ratio of the probability of getting the number 5 to the probability of not getting the number 5.

Thus, we conclude that the odds against an event are the opposite of the odds in favor of the event.

Let us understand the meaning of "odds" with an example. This will also lead us on to the formula written below it.

What are the odds in favor of getting the number 5 on rolling a number cube?

First, calculate the probability of getting the number 5 on rolling a number cube. We use the formula for theoretical probability to calculate it.
P(5) = "Number of favorable outcomes"/"Total number of outcomes"
P(5) = 1/6
Now calculate the probability of not getting the number 5 on rolling the number cube. This is the complement of the probability of getting 5, thus,
P("not 5") = 1 - P(5) = 1 - 1/6 = 5/6
Now, odds in favor of getting the number 5 on rolling a number cube are given by the ratio of the above probabilities.
"Odds in favor of getting 5" = (P(5))/(P("not 5"))
"Odds in favor of getting 5" = (1/6)/(5/6) = 1/5"
Thus, the odds in favor of getting the number 5 on rolling a number cube are 1/5.

From the above example, we can conclude that the formula for odds in favor of an event can be written as follows:
"Odds in favor of an event E" = (P("E"))/(P("not E"))
Similarly, since odds against an event is the ratio of the probability of not happening of that event to the probability of happening of that event, therefore its formula is
"Odds against of an event E" = (P("not E"))/(P("E"))

Thus, as stated previously, we can say that the odds against an event are the complete opposite of the odds in favor of it. In other words, the odds against an event are the reciprocal of the odds in favor of an event. For example, if the odds against an event are 1/2, then the odds in favor of that event are 2/1.

The above example leads us to another conclusion: Since the odds against or in favor of an event are the ratio of probabilities and not the probabilities themselves, therefore they can be greater than 1as opposed to probability of an event (recall that the probability of an event can not be greater than 1).

Further, if we use the formula for theoretical probability, we can write
P("E") = "Number of favorable outcomes"/"Total number of outcomes"
and,
P("not E") = "Number of outcomes that are not favorable"/"Total number of outcomes"
If we place the above two formulas for P(E) and P(Not E) in the formula for odds in favor of an event E, we get
"Odds in favor of event E" = (P("E"))/(P("Not E"))
"Odds in favor of event E" = ("Number of favorable outcomes"/"Total number of outcomes")/("Number of outcomes that are not favorable"/"Total number of outcomes")
Simplifying, we get,
"Odds in favor event E" = "Number of favorable outcomes"/"Number of outcomes that are not favorable"
The above formula is extremely useful in calculating the odds in favor of an event. It helps you calculate the odds in favor of an event without calculating the probability of its happening or not happening.

Since the odds against an event are the reciprocal of the odds in favor of it, thus, we can also write the formula for odds against an event as follows:
"Odds against an event E" = "Number of outcomes that are not favorable"/"Number of favorable outcomes"

Solved Examples

1. There are five red, four blue and three white marbles in a bag. What are the odds against and in favor of getting a red marble on drawing one marble from the bag?

There are five red marbles in the bag, therefore the number of favorable outcomes for drawing a red marble is 5. The other 7 marbles are not red. Thus the number of outcomes that are not favorable are 7. Thus,
"Odds against getting a red marble" = "Number of outcomes that are not favorable"/"Total number of outcomes"
"Odds against getting a red marble" = 7/5
Since the odds in favor of an event are the opposite of the odds against it, therefore,
"Odds in favor of getting a red marble" = 1/"Odds against it" = 5/7

2.  A number cube is rolled. What are the odds against getting a number greater than 4 on it? What are the odds in favor?

There are two numbers, 5 and 6, greater than 4 on a number cube. Thus, the number of favorable outcomes is 2. The other four numbers on the number cube (1 to 4) are not greater than 4. Thus the number of outcomes that are not favorable is 4.
"Odds against getting a number greater than 4" = "Number of outcomes that are not favorable"/"Number of outcomes that are favorable"
"Odds against getting a number greater than 4" = 4/2 = 2/1
Since the odds in favor of an event are the opposite of the odds against an event, therefore

"Odds in favor of getting a number greater than 4 = 1/2"

3. One card is drawn from a deck of fifty two playing cards. What are the odds in favor of getting a red King?

There are two red Kings in a standard deck of fifty two cards. Thus the number of favorable of favorable outcomes is 2. The other 50 cards are not red Kings. Thus the number of outcomes that are not favorable are 50.
"Odds in favor of getting a red King" = "Number of favorable outcomes"/"Total number of outcomes"
"Odds in favor of getting a red King" = 2/50= 1/25

4. It rained on four out of five consecutive days in a week. What are the odds against raining on the sixth day?

Since it rained on four days, number of favorable outcomes for raining = 4
Since it did not rain on one day, therefore number of non-favorable outcomes = 1
Applying the formula for "odds against",
"Odds against raining" = "Number of non-favorable outcomes"/"Number of favorable outcomes" = 1/4

5. Team A won three of the four matches against Team B. What are the odds in favor of team A winning the fifth match?

Since team A won three matches, therefore number of favorable events for wining a match of team A = 3
Since team A did not win one match out of four, therefore number of non-favorable outcomes for winning a match of team A = 1
Applying the formula for "odds in favor",
"Odds in favor of team A winning the match" = "Number of favorable outcomes"/"Number of non-favorable outcomes" = 3/1 = 3