| Odds against is a ratio of the probability of A not occurring to the probability of A occurring. |
A typical method of calculating this would be P(Ã) to P(A), where P(Ã) is the probability of A not happening and P(A) is the probability of A happening. These might possibly be expressed as a fraction, but beware that odds are not probability. Notice that if these probabilities are expressed as fractions, the denominators of both could be the same and could represent the number of outcomes in our sample space, if unreduced.
| The odds in favor or odds for is the "reciprocal" of the odds against: P(A) to P(Ã). |
Odds against or odds in favor [of] are sometimes left unreduced, but are typically reduced with a "numerator" of 1. They are most commonly expressed in the form a:b or a to b. When one says you have a 50:50 chance of getting heads, this is a typical statement of odds, but can also be interpretted as saying you have a 50% chance of winning and a 50% chance of losing.
Example: Baseball
and most such sports commonly use probabilities and
not odds. Batting average, or hits per at bat, would be a typical example.
Given a batting average of 0.250 and a third of those hits being for extra
bases (0.083), what are the odds against getting a hit, getting an
extra base hit, or the odds of a hit being for extra bases?
Solution:The odds against getting a hit would be 0.750 to 0.250
which reduces to 3 to 1.
The odds of [/against] getting an extra base hit would be
0.917 to 0.083 or, using the original information or three significant
figures, 11 to 1.
The odds of [/against] a hit being for extra bases would be
0.167 to 0.083 or 2 to 1.
Odds are commonly used in gambling. Some common applications might be horse-racing, the lottery, or the roulette wheel. Assume a typical American-style roulette wheel numbered 00 and 0 to 36. (European-style roulette wheels do not have the 00 and thus give better odds.) We would say the odds against selecting the correct winning slot would be 37 to 1. The odds in favor would be 1 in 37.
Although the use of odds makes it easier to deal with money exchanges resulting from gambling, odds are awkward to use in calculations. That is why they are converted into probabilities when, for example, applying the multiplication rule for combining independent events. The odds against an event represent the ratio of net profit to the amount bet.
Example: What is the event probability and
net profit for a bet which pays 50:1?
Solution: More than likely these are odds against
since it was not specified. If you bet $2, your net profit would be $100.
That is, you would collect a total of $102. The corresponding probabilities
would be 1/51.
Example: What would be the odds against rolling a total of 4 using
three regular six-sided dice.
Solution:
There is but one way to roll a three (all dies showing ones),
but there are three ways (any die shows two, the others show one)
of getting the total of four.
There are 63 or 216 different outcomes.
The odds against would be 216-3 to 3 = 213 to 3.
This would be more typically expressed as 71 to 1.
To say one has long odds would mean it is unlikely (say, 10 to 1 or 100 to 1).
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