A protein that has the objective of assisting pig growth is given to pigs in their lunch. A researcher observes that few pigs may have developed tumours as a result of their eating which comprises the protein. The researcher who added the protein discovers that for a random pig that gets tumours - there is a probability ranging from p=1/10, when a pig has a standard lunch to p=(1/2) when given the lunch that has the protein present. The researchers' investigation has 100 pigs, where the number of pigs that are given the lunch with the protein present is unknown (I refer to this as n).
My question that I previously solved is: A second researcher randomly chooses a pig from the 100 pig cohort. If the pig has tumours, find the probability that the pig had been given lunch where the protein is present.
My working out:
I have established that: Event A = Has tumours Event B = lunch with protein
P(B) = n100. P(B') = 100−n100. P(A|B) = 0.5. P(A|B') = 0.1
I then apply Bayes Theorem:
Algrebraic manipulation of:
P(B|A) = 0.5n/100((0.5n/100)+(100−n)/(1000) = 5n4n+100
Question: Find the largest number of pigs that ought to be given the lunch with the protein present if the second researcher wants the above probability to be smaller than 0.5.
My working out:
5n4n+100 < 0.5
5n < 0.5(4n+100)
5n < 2n + 50
3n < 50
n < 503 = 16.7
And so n=16.
Is this correct, I feel that this is too simple however perhaps that is from doing questions of a similar nature.
Any help is welcome. Thanks in advance.