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Why Do We Multiply the Probabilities of Independent Events?

Date: 10/15/2009 at 22:05:25
From: Hannah
Subject: Why use multiplication to find the probability of two events

I would like to know the reasoning behind this topic.

P(A and B)= P(A)x P(B)

Why do we have to multiply the two probabilities, why can't we add
them together?  Thank you very much.




Date: 10/15/2009 at 23:38:02
From: Doctor Peterson
Subject: Re: Why use multiplication to find the probability of two events

Hi, Hannah.

It may be clearer to you if you think of probability as the fraction
of the time that something will happen.  If event A happens 1/2 of the
time, and event B happens 1/3 of the time, and events A and B are
independent, then event B will happen 1/3 of the times that event A
happens, right?  And to find 1/3 of 1/2, we multiply.  The probability
that events A and B both happen is 1/6.

Note also that adding two probabilities will give a larger number than
either of them; but the probability that two events BOTH happen can't
be greater than either of the individual events.  So it would make no
sense to add probabilities in this situation.

Does that help?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/02/2016 at 01:16:54
From: Mujarie
Subject: Why Do We Multiply the Probabilities of Independent Events?

Sir, can you please explain more your answer in the question above?

I don't understand this part:

"If event A happens 1/2 of the time, and event B happens 1/3 of the time, 
and events A and B are independent, then event B will happen 1/3 of the 
times that event A happens ..."

Sorry, kindly explain that more.

Hoping for your quick response. Thank you.

Date: 02/02/2016 at 08:22:09
From: Doctor Peterson
Subject: Re: Why Do We Multiply the Probabilities of Independent Events?

Hi, Mujari.

Let's take a concrete example. 

Suppose, as you say, we have event A with probability 1/2, and event B 
with probability 1/3, and they are independent. Then suppose that there 
are 60 equally likely outcomes represented by these dots:

   o o o o o o o o o o
   o o o o o o o o o o
   o o o o o o o o o o
   o o o o o o o o o o
   o o o o o o o o o o
   o o o o o o o o o o

Now suppose that the A's represent the outcomes in event A (1/2 of 
all outcomes):

   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o

Now, among the 30 outcomes in which A occurred, B occurs in 1/3 of them 
(10); and the same is true among outcomes in which A did not occur. I'll 
use B to indicate that ONLY B occurred, and X to indicate that BOTH A and 
B occurred:

   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o
   A A A A A o o o o o
   X X X X X B B B B B
   X X X X X B B B B B

How many times did BOTH occur? 1/3 of the 1/2, which is 1/6 of all 
outcomes; that is 1/6 of my 60 outcomes, which is 10, as illustrated in 
the lower left-hand region, above.

So the probability that A and B occur is 1/2 * 1/3 = 1/6 -- that is, 10/60 
in the picture.

Does that help at all?

- Doctor Peterson, The Math Forum
  

Date: 02/03/2016 at 02:41:35
From: Mujarie
Subject: Thank you (Why Do We Multiply the Probabilities of ...?)

Thank you so much for answering my question.
Associated Topics:
High School Probability
Middle School Probability

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