Odds and Probability

A brief explanation and the differences between odds and probability.

Definition of Odds:

Odds in probability of a particular event, means the ratio between the number of favorable outcomes to the number of unfavorable outcomes.

Odds in favor and odds in against – probability:

Odds in favor:

Odds in favor of a particular event are given by Number of favorable outcomes to Number of unfavorable outcomes.

                             Number of favorable outcomes
            ^{P(A) =}     Number of unfavorable outcomes

For example;

Find the odds in favor of throwing a die to get “3 dots”.

Solution:

Total number of outcomes in throwing a die = 6

Number of favorable outcomes = 1

Number of unfavorable outcomes = (6 - 1) = 5

Therefore, odds in favor of throwing a die to get “3 dots” is 1 : 5 or 1/5

Odds against:

Odds against is given by Number of unfavorable outcomes to number of favorable outcomes.

                           Number of unfavorable outcomes
            ^{P(A) =}      Number of favorable outcomes   

For example;

Find the odds in against of throwing a die to get “3 dots”.

Solution:

Total number of outcomes in throwing a die = 6

Number of favorable outcomes = 1

Number of unfavorable outcomes = (6 - 1) = 5

Therefore, odds in against of throwing a die to get “3 dots” is 5 : 1 or 5/1

Then,

Probability of the event=

                                Number of favorable outcomes                                
    Number of favorable outcomes + Number of unfavorable outcomes

Worked-out Problems on Odds and Probability:

1. If odds in favor of X solving a problem are 4 to 3 and odds against Y solving the same problem are 2 to 6.

Find probability for:

(i) X solving the problem

(ii) Y solving the problem

Solution:

Probability of the event =

                                Number of favorable outcomes
    Number of favorable outcomes + Number of unfavorable outcomes

Given odds in favor of X solving a problem are 4 to 3.

Number of favorable outcomes = 4

Number of unfavorable outcomes = 3

(i) X solving the problem

P(X) = P(solving the problem) = 4/(4 + 3)

= 4/7

Given odds against Y solving the problem are 2 to 6

Number of favorable outcomes = 6

Number of unfavorable outcomes = 2

(ii) Y solving the problem

P(Y) = P(solving the problem) = 6/(2 + 6)

= 6/8

= 3/4

2. What is the difference between odds and probability?

Solution:

The difference between odds and probability are:

Odds of an event are the ratio of the success to the failure.

                           success
            ^{Odds =}     Failures

Probability of an event is the ratio of the success to the sum of success and failure.

                                     success
            ^{Odds =}     (Success + Failures)

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