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n insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims. Calculate the probability that a claim submitted to the insurance company includes operating room charges.

Respuesta :

Answer:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

Step-by-step explanation:

First we can define some notation useful:

C ="represent the event of incurring in operating charges"

R= represent the event of emergency rooms charges"

For this case we are interested on P(C) since they want "the probability that a claim submitted to the insurance company includes operating room charges."

We have some probabilities given:

[tex] P(R') = 0.25 , P(C \cup R) =0.85[/tex]

Solution to the problem

By the complement rule we have this:

[tex] P(R') = 0.25 =1-P(R)[/tex]

[tex] P(R) = 1-0.25 = 0.75[/tex]

Since the two events C and R are considered independent we have this:

[tex]P(C \cap R) = P(C) *P(R)[/tex]

Now we can use the total probability rule like this:

[tex] P(C \cup R) = P(C) + P(R) - P(R)*P(C)[/tex]

And if we replace we got:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

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