Use Venn diagrams to find out how many times we are adding each region.
A
a
P(Aor BorC) = P(A) + P(B) + P(C)
B
b
P(A or BorC) = P(A) + P(B) + P(C)-P(A and B) - P(B and C) - P(A and C) + P(A and BandC)
Practice makes perfect
a
We are asked to write a general rule to find P(Aor B orC) for disjoint events A, B, and C. Let's look at the Venn diagram of three disjoint events.
We can see that these events do not share any area in the Venn diagram. Therefore, similar to the probability of Aor B, we can find the probability of Aor BorC by adding each individual probability.
P(Aor BorC) = P(A) + P(B) + P(C)
b
Now we are asked to write a general rule to find P(Aor B orC) for overlapping events A, B, and C. Let's look at the Venn diagram of three overlapping events.
We can see that there are several overlapping sections. To find the probability of P(Aor B orC), we need to add the probabilities of each region of the diagram only once. We know how to find the probability for two overlapping events.
P(AorB) = P(A) + P(B) - P(A and B)
Let's begin to write our rule by extending this previous knowledge to be for three overlapping events. We will do so by adding the probability of each event. Then, we subtract from the sum the probabilities of each of the intersections.
P(A) + P(B) + P(C) -
P(A and B)-P(B and C)
-P(A and C)
Let's add these probabilities in a Venn diagram to check if we have succeeded in writing the rule. Each number in the diagram represents the number of times that we have added each region.
We can see that P(A and BandC) has not been added yet. Therefore, we will add this probability to complete our rule.
P(A or BorC)
=
P(A) + P(B) + P(C)
-P(A and B)-P(B and C) -P(A and C) +
P(A and BandC)
Finally, let's add this to our diagram.
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var txtB = b.txt(0,4.4,"+P(B)",{color:"transparent"});
var txtC = b.txt(1.6,4.4,"+P(C)",{color:"transparent"});
var txtAB = b.txt(-3,3.7,"-P(A\\text{ and }B)",{color:"transparent"});
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/* setting some values */
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var borderColor = 'gray';
/* setting the colors of the regions
use 'transparent' if no color is needed */
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var optionsB = {fillColor:'transparent',fillOpacity:op,borders:{visible:false}};
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var optionsAB = {fillColor:'transparent',fillOpacity:op,borders:{visible:false}};
var optionsBC = {fillColor:'transparent',fillOpacity:op,borders:{visible:false}};
var optionsCA = {fillColor:'transparent',fillOpacity:op,borders:{visible:false}};
var optionsABC = {fillColor:'transparent',fillOpacity:op,borders:{visible:false}};
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var valueB = ;
var valueC = ;
var valueAB = ;
var valueBC = ;
var valueCA = ;
var valueABC = ;
var valueABCout = ;
/* setting the names of the circles */
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var textB = 'B';
var textC = 'C';
/* No need to change anything beyond this point.
The rest of the code uses the values set above */
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