The multiplication rule is used to find the probability of two events happening at an equivalent time (this is additionally one among the AP Statistics formulas). There are two multiplication rules - the all total multiplication rule formula is written as P(A ∩ B) = P(A) P(B|A) and the specific multiplication rule is P(A and B) = P(A) * P(B). P(B|A) it means that the probability for an event as long as the B event has occurred”.
The specific multiplication rule, P(A and B) = P(A) * P(B), is merely valid if the 2 events are independent. In other words, it only works if one event doesn't change the probability of the opposite event.
Examples of Independent Events
Owning a cat and getting a weekly paycheck.
Finding a parking lot and having a coin for the meter.
Buying a book after that buying a coffee.
In some cases, the primary event happening impacts the probability of the second event. We call these dependent events. In other cases, the primary event happening doesn't impact the probability of the seconds. We call these independent events.
Multiplication Rule Probability: Using the Specific Rule
Using the precise multiplication rule formula is extremely straightforward. Just multiply the probability of the primary event by the second. for instance, if the probability of event A is 2/9 and therefore the probability of event B is 3/9 then the probability of both events happening at an equivalent time is (2/9)*(3/9) = 6/81 = 2/27.
Multiplication Rule Probability: Using the General Rule
This rule is often used for any event (they are often independent or dependent events). You continue to need to multiply two numbers, but first, you've got to use a touch logic to work out the second probability before multiplying.
Sample Problem: A bag contains 6 black marbles and also 4 blue marbles. Two of the marbles are drawn from the bag, without any replacement. What will be the probability that both of the marbles will be of colour blue?
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Step 1: Label your events A and B. Let A be the event that the marble 1 is supposed to be blue and let B be the event that marble 2 is supposed to be blue.
Step 2: Find out the probability for A. There are ten marbles inside the bag, so the probability for drawing a blue marble is 4/10.
Step 3: Find out the probability for B. There are nine marbles inside the bag, so the probability for selecting a blue marble (P B|)A is 3/9.
Step 4: Multiply Step 2 and three together: (4/10)*(3/9) = 2/15.
Multiplication Rule of Probability Statement and Proof
We know that the contingent probability of event A as long as B has occurred is denoted by P(A|B) and is given by:
P(A|B)= P(A∩B) /P(B)
P(A∩B) = P(B)×P(A|B) ……………………………………..(1)
P(B|A)= P(B∩A) /P(A)
Where P(A) ≠ 0.
P(B∩A) = P(A)×P(B|A)
Since, P(A∩B) = P(B∩A)
P(A∩B) = P(A)×P(B|A) ………………………………………(2)
From (1) and (2), we get:
P(A∩B) = P(B)×P(A|B) = P(A)×P(B|A) where,
P(A) ≠ 0,P(B) ≠ 0.
The above results are referred to as the multiplication rule of probability.
For the independent events which are A and B, P(B|A) = P(B). The equation (2) is often modified into,
P(A∩B) = P(B) × P(A)
Multiplication Theorem in Probability
We have already learned about the multiplication rules we follow in the probability, given below;
P(A∩B) = P(A)×P(B|A) ; if P(A) ≠ 0
P(A∩B) = P(B)×P(A|B) ; if P(B) ≠ 0
Let’s learn about the multiplication theorems for the independent events for A and B.
If A and B are two independent events for a random experiment, then the probability of simultaneous occurrence of two independent events are going to be adequate to the merchandise of their probabilities. Hence,
P(A∩B) = P(A).P(B)
Now, from multiplication rule we know;
P(A∩B) = P(A)×P(B|A)
Since A and B are independent, therefore;
P(B|A) = P(B)
Therefore, again we get;
P(A∩B) = P(A).P(B)