# Conditional Probability at it's finest

• Somefantastik
In summary, conditional probability is the likelihood of an event occurring given that another event has already occurred. It differs from regular probability in that it takes into account the relationship between two events. An example of conditional probability is flipping a coin twice, where the probability of the second flip is affected by the outcome of the first flip. It is useful in various fields such as medicine, finance, and engineering. Some common misconceptions about conditional probability include its limitation to only two events and its applicability only to independent events.
Somefantastik
In an election, candidate A receives n votes and candidate B receives m votes, where n>m. Assume that in the count of the votes all possible orderings of the n+m votes are equally likely. Let Pn,m denote the probability that from the first vote on A is always in the lead. Find Pn,m.

[solution]
Pn,m = P(A gets n votes, B gets m votes | last vote to A)*(n/n+m) + P(A gets n votes, B gets m votes | last vote to B) * (m/m+n)

= Pn-1,m * (n/n+m) + Pn,m-1*(m/m+n)

From the formula, we have

Pn,1 = Pn-1,1 *(n/n+1) + Pn,0*(1/n+1) = (n/n+1)*Pn-1,1 + 1/(n+1)
P1,1 = 0

=> Pn,1 = (n-1)/(n+1)

and

Pn,2 = Pn-1,2*(n/n+2) + Pn,1*(2/2+n) = Pn-1,2*(n/n+2) + 2*(n-1)/(n+2)
P2,2 = 0

==> Pn,2 = (n-2)/(n+2)

Please explain this solution to me. The stuff in red is where I got lost.

This is a common method for 'random walk' type problems which are solved by difference equations. Pn,m is called 'steady state probability' and the corresponding equation is called ballance equation of transition. If you understand the first expression of Pn,m then it will not be difficult to understand how Pn-1,m and Pn,m-1 are eleminated. Go through the method of solving difference equations in any textbook. Hope this is not a homework.

Last edited:
Yes it's a homework but it has already been turned in and graded. Any more responses are appreciated.

Somefantastik said:
Pn,m = Pn-1,m * (n/n+m) + Pn,m-1*(m/m+n)
Since this wasn't in red I assume you have no problem with this.

From the formula, we have
Pn,1 = Pn-1,1 *(n/n+1) + Pn,0*(1/n+1) = (n/n+1)*Pn-1,1 + 1/(n+1)
The first part of this statement results directly from substituting m=1 into the generic recursive relationship. If candidate B does not receive any votes then candidate A will always be in front. Thus Pn,0=1 for all n>0. The latter equality arises from this fact.
From this, the latter equality need this to make the latter part of the above statement.
P1,1 = 0
If candidate B receives as many or more votes than candidate A there will be some point at which candidate A is not in front. Thus Pn,m=0 for all n<=m.
=> Pn,1 = (n-1)/(n+1)
The recursive relationship suggests there might be a direct expression of the form
Pn,1 = f(n)/(n+1). Using this form in the recursive relationship,
f(n)/(n+1) = (n/(n+1))*(f(n-1)/n) + 1/(n+1)
from which
f(n) = f(n-1) + 1
In other words, f(n) = n+k where k is some constant. Since P1,1=0, f(1)=0 and k=-1, or
Pn,1 = (n-1)/(n+1)

You can verify this with recursion. The expression is obviously correct for n=1. Assuming its correct for n-1, you should be able to prove it is true for n.

The case m=2 is similar.

## 1. What is conditional probability?

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It takes into account the relationship between two events and can be calculated using the formula P(A|B) = P(A∩B) / P(B), where P(A|B) is the conditional probability of A given B, P(A∩B) is the joint probability of A and B, and P(B) is the probability of B.

## 2. How is conditional probability different from regular probability?

Regular probability calculates the likelihood of an event occurring without taking into account any other information, while conditional probability takes into account the relationship between two events. In other words, regular probability is based on a single event, while conditional probability is based on multiple events.

## 3. What is an example of conditional probability?

An example of conditional probability is flipping a coin twice. The probability of getting heads on the first flip is 1/2, but the probability of getting heads on the second flip given that the first flip was heads is now 1/2 * 1/2 = 1/4. This is because the two events are related and the outcome of the first flip affects the probability of the second flip.

## 4. How is conditional probability useful in real life?

Conditional probability is useful in real life in various fields such as medicine, finance, and engineering. For example, in medicine, conditional probability can be used to calculate the likelihood of a patient having a certain disease given their symptoms. In finance, it can be used to predict the risk of an investment based on market conditions. In engineering, it can be used to assess the reliability of a system based on the performance of its individual components.

## 5. What are some common misconceptions about conditional probability?

One common misconception about conditional probability is that it can only be used for two events. In reality, it can be applied to multiple events in a chain, as long as the relationship between each event is known. Another misconception is that conditional probability can only be used for independent events. However, it can also be used for dependent events as long as the relationship between them is understood.

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