All occurrences of A and B in the sample space S are represented by the set AB. Because of this, we may say that the combination of events A and B is denoted by the symbol (AB). The letters AB stand for the letters A and B together. The probability of event AB may be estimated using conditional probability characteristics.
Exactly how does one multiply probabilities?
Using the multiplication rule of probability, we can calculate the chance that events A and B will occur together as the product of the probability that event B will occur and the conditional probability that event A will happen, given that event B occurs. For more examples and questions using percentages, check out their webpage.
If occurrences A and B are dependent on one another, then the probability that they will co-occur may be written as follows:
P(A ∩ B) = P(B) . P(A|B)
Assuming A and B are independent in an experiment, the following formula gives the probability that they will occur at the same time.
P(A ∩ B) = P(A) . P(B)
Proof
This is the definition of P(A|B), the conditional probability of event A given the occurrence of event B.
P(B)=0 if and only if
This means that P(A|B) (the probability of A occurring given B) is equal to P(B|A) (the probability of B occurring given A). ……………………………………..(1)
Given that P(A) = 0,
The probability that B is less than or equal to A is equal to the sum of the probabilities of A and P(B|A).
Therefore, P(AB) = P (BA)
That is, if we have A and B in the first set of observations, then the likelihood of A given B is equal to the product of the probabilities of A and B in that set. ……………………………………… (2)
Consequently, if we consider (1) and (2),
To elaborate, P(AB) = P(A)P(B|A) if and only if P(B)P(A|B) = P(A)P(B|A).
P(A) ≠ 0,P(B) ≠ 0.
This is known as the “multiplication law of probability.”
If events A and B are unrelated, then P(B|A) = P. (B). By making a change to Eq. (2), we get
P(AB) = P(B) P is the formula for the probability of A occurring given the likelihood of B occurring (A)
Combining the results of the probabilistic theorem with the product theorem
Many of the multiplication rules used in probability, such as
In the event when P(A) = 0, P(A) P(B|A) = 0.
In the event when P(A|B) = 0, P(AB) = P(B)P(A|B).
Theorems for multiplying two independent events, A and B, will be investigated in this article.
The probability of an event A and an event B happening simultaneously in a random experiment is proportional to the product of those two probabilities. Hence,
For every two sets A and B, P(AB) = P(A) (A).
P(B)
What we can glean from the multiplication rule is this:
That is, if we have A and B in the first set of observations, then the likelihood of A given B is equal to the product of the probabilities of A and B in that set.
For the reason that both A and B may function independently, it follows that
P(B|A) = P(B) (B)
This time around, we arrive at;
Likewise, P(AB) = P(A) (A).
P(B)
thus, proven.
Multiplying Probabilities: A Worked Example
In the first illustration, an urn contains ten blue and twenty red balls. Two balls are selected at random from a bag and used up before being replaced. Is it possible that two red balls will be selected at random?
We may answer this by assuming that the probability of selecting a red ball as the first and second balls are A and B, respectively. To do this, we must first decide if P(AB) or P (AB).
Probability of a red ball being drawn first = 20/30 = P(A).
There are only 19 red balls and 10 blue balls left in the bag at this point. One such example of conditional probability is the likelihood of drawing a red ball based on the colour of the ball drawn before.
Because of this, we can write out the formula for the probability of B given A:
P(B|A) = 19/29
Through application of the theorem of multiplication,
Assume that P(AB) = P(A) P(B|A).
Probability via Adding Up
The probability of two events happening is equal to the difference in the probabilities of those events occurring, minus one, in accordance with the addition rule.
There exists a mathematical expression for the probabilistic rule of addition, and it reads as follows:
A’s chance of happening if B occurs is equal to (A’s chance of occurring if B occurs minus) A’s chance of occurring if B occurs.
