If Ω is finite and we have a discrete uniform probability law, and if B≠∅, then the conditional probability law on B, given that B occurred, is also discrete uniform.
True, because all outcomes are equaly likely to hapen and have a probability of 1.
To show that the conditional probability law on B, given that B occurred, is also discrete uniform, we need to verify two conditions:
1. The conditional probabilities must sum to 1.
2. The probabilities of the individual outcomes must be equal.
Let's go through these steps to prove that the conditional probability law is discrete uniform:
Step 1: Conditional Probabilities Sum to 1
Since B≠∅, we know that B occurred. Therefore, the sample space Ω is now restricted to B. The sum of probabilities in any discrete probability law must be equal to 1, so we need to calculate the sum of conditional probabilities on B.
The sum of probabilities on B, given that B occurred, is given by:
P(B|B) + P(B' | B) = 1 + P(B' | B)
Since B and B' are mutually exclusive events, P(B' | B) is equal to 0. Therefore:
P(B|B) + P(B' | B) = 1 + 0 = 1
This shows that the sum of conditional probabilities on B, given that B occurred, is indeed equal to 1.
Step 2: Probabilities of Individual Outcomes are Equal
We need to show that the probabilities of the individual outcomes in B, given that B occurred, are equal.
Let's assume there are m outcomes in B. Since we have a discrete uniform probability law initially, the probability of each outcome in Ω is 1/n, where n is the total number of outcomes in Ω. Now, considering the restriction B, the probability of each outcome in B, given that B occurred, would be P(Bi | B) = P(Bi and B) / P(B), where Bi represents the individual outcomes in B.
Since B occurred, the denominator P(B) is equal to 1. Hence, P(Bi | B) = P(Bi and B).
Since the original probability law was discrete uniform, we have:
P(Bi) = 1/n for all i
Now let's calculate P(Bi and B). Since B occurred, only the outcomes in B are valid. Since B ≠ ∅, we can say that Prob(Bi and B) = 1/m for all i, where m is the total number of outcomes in B.
Therefore, we have:
P(Bi | B) = P(Bi and B) / P(B) = (1/m) / 1 = 1/m
This shows that the probabilities of the individual outcomes in B, given that B occurred, are equal and given by 1/m, where m is the total number of outcomes in B.
Thus, we have shown that the conditional probability law on B, given that B occurred, is also discrete uniform.
To prove that the conditional probability law on B, given that B occurred, is also discrete uniform, we need to show that all the outcomes in B have equal probabilities when conditioned on B.
Let's assume that Ω is a finite sample space, and we have a discrete uniform probability law on Ω. This means that every outcome in Ω has an equal chance of occurring.
Now, let's consider event B ⊆ Ω such that B is non-empty (B≠∅). The conditional probability law on B, denoted as P(· | B), is defined as the probability of an event A occurring, given that B has already occurred.
To calculate the conditional probability of any outcome in B, given that B occurred, we need to divide the probability of that outcome by the probability of event B. Since B is non-empty, its probability is greater than zero (P(B) > 0).
Since we have a discrete uniform probability law on Ω, every outcome in Ω has an equal chance of occurring. Therefore, the probability of any outcome in B is equal to the probability of any other outcome in B. Let's call this common probability value p.
Now, let's calculate the conditional probability of an outcome in B, given that B occurred. Using the definition of conditional probability, we have:
P(A | B) = P(A ∩ B) / P(B)
Since P(A ∩ B) represents the joint probability of A and B occurring, and B is given to have occurred, we can rewrite it as:
P(A ∩ B) = P(A | B) * P(B)
Substituting the value of P(A ∩ B) into the conditional probability formula, we get:
P(A | B) = (P(A | B) * P(B)) / P(B)
Simplifying this equation, we see that P(A | B) = P(A).
This implies that the conditional probability of any outcome in B, given that B occurred, is equal to p. Since every outcome in B has the same probability value p, we can conclude that the conditional probability law on B, given that B occurred, is also discrete uniform.
In summary, if Ω is finite and we have a discrete uniform probability law, and if B≠∅, then the conditional probability law on B, given that B occurred, is also discrete uniform because every outcome in B has equal probabilities when conditioned on B.