Question

Check my work, please
During a visit to a primary care physician's office, the probability of having neither lab work nor referral to a specialist is 0.21. Of those coming to that office, the probability of having lab work is 0.41 and the probability of having a referral is 0.53.
What is the probability of having both lab work and a referral?

Answer 1

Let A :={x | x has lab work} --> P(A) = 0.41
B:= {x | x has referral} --> P(B) = 0.53
\(P(A\cap B)' = 0.21\)

Answer 2

--> \(P (A \cap B ) = 0.78\), right?

Answer 3

0.78 looks very huge

Answer 4

--> \(P (A\cup B) = 0.78\)

Answer 5

\(P (A\cap B) = P(A) +P(B) - P( A\cup B) \\0.41+0.53-0.78=0.16\)

Answer 6

Am I right?

Answer 7

\[P(A'\cap B') =P(A\cup B)'= 0.21\]

Answer 8

\[\large P(A\cup B) = 1-P(A'\cap B') = 1-0.21 = 0.79\]

right ?

Answer 9

the final answer should be 0.15

just pointing out an arithmetic error :P

Answer 10

Thank you, I see my mistake now. :)

Answer 11

i would have done this using venn diagram to visualize
but i agree manipulating like this without using venn diagrams is very important to logically see the relationship