Odds and Probability || Formula for calculating odds

In this chapter, we will discuss the concept of odds and differentiate it with the concept of probability.

Understand that both probability and odds are the expressing the likelihood of an event but in different way.

What are Odds ?

Odds tells the likelihood of happening an event against non favorable outcome.

The formula for odd is given as;

\mathtt{Odds\ =\ \frac{Number\ of\ Favorable\ outcomes}{Number\ of\ Non\ Favorable\ outcome}} \\\ \\

For any analyst, the odds shows the ratio of number of outcomes in our favor to the outcomes not in favor.

Let us understand the concept with an example.

Suppose you went to a casino for betting. In the premises, the host welcomes and introduce you to a game. In the game, a dice will be rolled once and you will win the prize if the outcome is number 5.

Now as a gambler, you have to calculate the chance of winning the game.

Let event A be the favorable and B be the non favorable outcome.

A = { 5 }
Number of favorable outcome = 1

B = {1, 2, 3, 4, 6}
Number of unfavorable outcome = 5

\mathtt{Odds\ =\ \frac{Number\ of\ Favorable\ outcomes}{Number\ of\ Non\ Favorable\ outcome}}\\\ \\ \mathtt{Odds\ =\frac{1}{5} \ \ \ or\ 1:5}\\ \\

Hence, there is 1 out of 5 chance of winning the game. This mean that the chance of loosing the game is more.

In percentage, the odd of game is expressed as;

\mathtt{odds( in\ percentage) =\frac{1}{5} \times 100\ =20\%}

It means that 20% outcomes are in favor and 80% outcomes are not in favor.

Difference between odds and probability

There is a slight difference in odds and probability.

In probability, we calculate the chance of happening an event against total outcomes. Whereas, odds tells the likelihood of an event against unfavorable outcome.

There is also slight difference in probability and odds formula;

\mathtt{Probability\ =\ \frac{Number\ of\ favorable\ outcome}{Total\ possible\ outcomes}}\\\ \\ \mathtt{Odds\ =\ \frac{Number\ of\ Favorable\ outcomes}{Number\ of\ Non\ Favorable\ outcome}}

Let us understand both the concept with example.

Probability of event = 2/3
It tells that out of total 3 outcomes, 2 outcomes are in our favor.

Odds of event =2 / 3
It tells that for the experiment, 2 outcomes are in favor and 3 are not in favor. Hence, odds are against us as there are less favorable outcomes.

Questions on Odds and Probability

Question 01
Consider the experiment of throwing a dice. Calculate the probability and odd of getting number less than four.

Solution
Writing the sample space for experiment.
S = {1, 2, 3, 4, 5, 6 }

Total number of outcomes = 6

Let A be the event of getting number less than 4.
A = { 1, 2, 3 }

Total favorable outcomes = 3

Total unfavorable outcomes = 6 – 3 = 3

Calculating probability;

\mathtt{P( A) \ =\ \frac{Number\ of\ favorable\ outcome}{Total\ possible\ outcomes}}\\\ \\ \mathtt{P( A) \ =\ \frac{3}{6} =\ \frac{1}{2}}\\\ \\ \mathtt{P( A) \ ( percentage) =\frac{1}{2} \times 100=50\%}

So, there is 50% chance of happening the given event.

Calculating Odds

\mathtt{Odds\ =\ \frac{Number\ of\ Favorable\ outcomes}{Number\ of\ Non\ Favorable\ outcome}}\\\ \\ \mathtt{Odds=\frac{3}{3} =\ 1:1}

Hence, the odds for event A is 1 : 1

Question 02
The odds for winning championship is given as 2 : 3. Calculate the probability of the event.

Solution
The formula for odds = Favorable outcome / unfavorable outcome

Comparing the values, we get;

Number of favorable outcomes = 2
Number of unfavorable outcomes = 3

Total Outcomes = 2 + 3 = 5

Now applying the probability formula;

\mathtt{P\ =\ \frac{Number\ of\ favorable\ outcome}{Total\ possible\ outcomes}}\\\ \\ \mathtt{P\ =\ \frac{2}{5}} \\\ \\

Hence, there is 2/5 probability of winning the championship.

Question 03
Two coins are tossed simultaneously. Calculate the odds of getting atleast one head.

Solution
Writing the sample space for the experiment.
S = {(HH), (HT), (TH), (TT) }

Number of total outcomes = 4

Let A be the event of getting atleast one head.
A = {(HH), (HT), (TH)}

Number of favorable outcome = 3

Number of unfavorable outcome = 4 – 3 = 1

Now calculating the odds;

\mathtt{Odds\ =\ \frac{Number\ of\ Favorable\ outcomes}{Number\ of\ Non\ Favorable\ outcome}}\\\ \\ \mathtt{Odds=\frac{4}{1} =\ 4:1}

Hence, the odd for above event is 4 : 1