Rules of Probability – Multiplication 7

Now that you have understood the addition rule and how it can be used to compute the probabilities of two events A and B, let’s now understand the concept of the multiplication rule. This rule is applicable only on independent events about which you have already learnt before. Let’s take a look at how you can compute the probabilities of two events A and B occurring simultaneously.

As discussed in the video as well as in the previous segments, when an event A is not dependent on event B and vice versa, they are known as independent events. The multiplication rule allows us to compute the probabilities of both of them occurring simultaneously, which is given as:

    P(A and B) = P(A)*P(B).

Now, this rule can be extended to multiple independent events where all you need to do is multiply the respective probabilities of all the events to get the final probability of all of them occurring simultaneously. For example, if you have four independent events A, B, C and D, then:

 P(A and B and C and D) = P(A)*P(B)*P(C)*P(D).

Comparison between Addition Rule and Multiplication Rule

Both the addition rule and the multiplication rule allow you to compute the probabilities of the occurrence of multiple events. However, there is a key difference between the two, which should help you to decide when to use which rule.

  • The addition rule is generally used to find the probability of multiple events when either of the events can occur at that particular instance. For example, when you want to compute the probability of picking a face card or a heart card from a deck of 52 cards, a successful outcome occurs when either of the two events is true. This includes either getting a face card, a heart card, or even both a face and a heart card. This rule works for all types of events. 
  • The multiplication rule is used to find the probability of multiple events when all the events need to occur simultaneously. For example, in a coin toss experiment where you toss the coin three times and you need to find the probability of getting three heads at the end of the experiment, a successful outcome occurs when you get a head in the first, second and third toss as well. This rule is used for independent events only
  • Also, in the addition rule, do you remember the P(A⋂B) that we used to compute the final value of P(A⋃B)? This value is exactly the same as the P(A and B) that we compute in independent events using the multiplication rule. You can go back and verify it for the same example shown in the video. There we had P(Heart Card) = P(H) = 13/52, P(Face Card) = P(F) = 12/52 and P(Heart Card and Face Card) = P(H⋂F) = 3/52. Now, as mentioned by the multiplication rule, you can see that P(H and F) = P(H)*P(F) = (13/52)*(12/52) = 3/52, which is the same as the value of P(H⋂F).

Now, answer the following questions to strengthen your concepts in the topics taught in the last two segments.

In the next segment, we will sumamrise all the concepts you learnt in this session.

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