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Marrrta [24]
3 years ago
8

The Venn Diagram below models probabilities of three events, A,B, and C.

Mathematics
2 answers:
Keith_Richards [23]3 years ago
4 0

Answer: dependent

Step-by-step explanation:

PSYCHO15rus [73]3 years ago
4 0
<h2>Answer:</h2>
  • The two events are independent.
<h2>Step-by-step explanation:</h2>

By the conditional property we have:

If A and B are two events then A and B are independent if:

                  P(A|B)=P(A)

                               or

                 P(B|A)=P(B)

( since,

if two events A and B are independent then,

P(A\bigcap B)=P(A)\times P(B)

Now we know that:

P(A|B)=\dfrac{P(A\bigcap B)}{P(B)}

Hence,

P(A|B)=\dfrac{P(A)\times P(B)}{P(B)}\\\\i.e.\\\\P(A|B)=P(A) )

Based on the diagram that is given to us we observe that:

Region A covers two parts of the total area.

Hence, Area of Region A= 72/2=36

Hence, we have:

P(A)=\dfrac{36}{72}\\\\i.e.\\\\P(A)=\dfrac{1}{2}

Also,

Region B covers two parts of the total area.

Hence, Area of Region B= 72/2=36

Hence, we have:

P(B)=\dfrac{36}{72}\\\\i.e.\\\\P(B)=\dfrac{1}{2}

and A∩B covers one part of the total area.

i.e.

Area of A∩B=74/4=18

Hence, we have:

P(A\bigcap B)=\dfrac{18}{72}\\\\i.e.\\\\P(A\bigcap B)=\dfrac{1}{4}

Hence, we have:

P(A|B)=\dfrac{\dfrac{1}{4}}{\dfrac{1}{2}}\\\\i.e.\\\\P(A|B)=\dfrac{2}{4}\\\\i.e.\\\\P(A|B)=\dfrac{1}{2}

Hence, we have:

P(A|B)=P(A)

         Similarly we will have:

P(B|A)=P(B)

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