Only 1 in 1000 adults is afflicted with a rare disease for which a diagnostic te
ID: 3256333 • Letter: O
Question
Only 1 in 1000 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that when an individual actually has the disease, a positive result will occur 99% of the time, whereas an individual without the disease will show a positive lest result only 2% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease? b. An individual has 3 different email accounts. Most of her messages, in fact 70%, come into account #1, whereas 20% come into account #2 and the remaining 10% into account #3. Of the messages into account #1, only 1% are spam, whereas the corresponding percentages for accounts #2 and #3 are 2% and 5%, respectively. What is the probability that a randomly selected message is spam?Explanation / Answer
5a:
we use Bayes' theorem:
P[disease]=0.001
p[+|disease]=0.99
p[+|not disease]=0.02
p[disease|+] = p[+|disease]*p[disease]/ p[+]
=p[+|disease]*p[disease]/ {p[+|disease]*p[disease] + p[+|not disease]*p[not disease]}
=0.99*0.001/{0.99*0.001+0.02*(1-0.001)}
=0.047
5b.
Need to compute P[Spam]:
{again we will use Bayes theorem
P[Spam]=P[Spam|account1]*P[account1]+P[Spam|account2]*P[account2]+P[Spam|account3]*P[account3]
=0.01*0.7 + 0.02*0.2 + 0.05*0.1
=0.016
=1.6%
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