Let's do the Quick Strep Test first (something that wasn't in the assignment), so as not to give you the false impression that medical tests are all highly inaccurate...

In most cases, sore throat (pharyngitis) in children and adolescents occurs as a result of a viral infection. About 15% of sore throats are caused by Streptococcus bacteria.

According to Y Gurol et al., "The sensitivity and the specifity of rapid antigen test in streptococcal upper respiratory tract infections", Int J Pediatr Otorhinolaryngol 2010:

Rapid antigen detection and throat culture results for group A beta hemolytic streptococci from outpatients attending to our university hospital between the first of November 2005 and 31st of December 2008 were evaluated retrospectively. [...]

Rapid antigen detection sensitivity and specificity were found to be 64.6% and 96.79%, respectively.

The three numbers we need are

• Prevalence -- Proportion of Relevant Population with the Condition
• Sensitivity -- Proportion of people with the Condition who test positive
• Specificity -- Proportion of people without the Condition who test negative

In the case of the Quick Strep Test, those three numbers are 15/100, 0.646, 0.9679. So we get:

Given Prevalence of 0.15, 15000 in 100,000 relevant people will have the condition.

Given Sensitivity of 0.646, 0.646*15000=9690 of those with the condition will test positive,
and 15000-9690=5310 of those with the condition will test negative.

100000-15000 = 85000 out of 1000000 relevant people will not have the condition.

Given Specificity of 0.968, 0.968*85000 = 82272 of the people without the condition will test negative.
and 85000-82272 = 2728 of those without the condition will test positive.

So a total of 9690+2728 = 12418 people will test positive, of which 9690 actually have the condition.

Therefore the odds that a person who tests positive actually has the condition is 9690/12418,
or about 78.0 in a hundred.

In the other direction, a total of 5310+82272 = 87582 people will test negative,
of which 82272 are actually free of the condition.

Therefore the odds that a person who tests negative is actually free of the condition is 82272/87582, or about 93.9 in a hundred.

Now let's do the Mammography for Breast Cancer case -- the sources cited in the assignment gave the Prevalence, Sensitivity, and Specificity as 0.6/100, 0.886, 0.968.

Given Prevalence of 0.006, 600 in 100,000 relevant people will have the condition.

Given Sensitivity of 0.866, 0.866*600=520 of those with the condition will test positive, and 600-520=80 of those with the condition will test negative.

100000-600 = 99400 out of 1000000 relevant people will not have the condition.

Given Specificity of 0.968, 0.968*99400 = 96219 of the people without the condition will test negative,
and 99400-96219 = 3181 of those without the condition will test positive

So a total of 520+3181 = 3701 people will test positive, of which 520 actually have the condition.

Therefore the odds that a person who tests positive actually has the condition is 520/3701,
or about 14.1 in a hundred

In the other direction, a total of 80+96219 = 96299 people will test negative,
of which 96219 are actually free of the condition.

Therefore the odds that a person who tests negative is actually free of the condition is 96219/96299,
or about 99.9 in a hundred

Now let's do one of the %fPSA screening for Prostate Cancer cases. The cited sources give the Prevalence for men 55-59 as 150/100,000, and the Specificity as 0.222 at Sensitivity 0.90.

Given Prevalence of 0.0015, 150 in 100,000 relevant people will have the condition.

Given Sensitivity of 0.900, 0.900*150=135 of those with the condition will test positive, and 150-135=15 of those with the condition will test negative.

100000-150 = 99850 out of 1000000 relevant people will not have the condition.

Given Specificity of 0.222, 0.222*99850 = 22167 of the people without the condition will test negative.
and 99850-22167 = 77683 of those without the condition will test positive

So a total of 135+77683 = 77818 people will test positive, of which 135 actually have the condition.

Therefore the odds that a person who tests positive actually has the condition is 135/77818,
or about 0.2 in a hundred

In the other direction, a total of 15+22167 = 22182 people will test negative,
of which 22167 are actually free of the condition.

Therefore the odds that a person who tests negative is actually free of the condition is 22167/22182, or about 99.9 in a hundred

The prevalence for men aged 75-79 is much higher, about 800 per 100,000. At that rate, things work out like this:

Given Prevalence of 0.008, 800 in 100,000 relevant people will have the condition.

Given Sensitivity of 0.900, 0.900*800=720 of those with the condition will test positive,
and 800-720=80 of those with the condition will test negative.

100000-800 = 99200 out of 1000000 relevant people will not have the condition.

Given Specificity of 0.222, 0.222*99200 = 22022 of the people without the condition will test negative.
and 99200-22022 = 77178 of those without the condition will test positive

So a total of 720+77178 = 77898 people will test positive, of which 720 actually have the condition.

Therefore the odds that a person who tests positive actually has the condition is 720/77898,
or about 0.9 in a hundred

In the other direction, a total of 80+22022 = 22102 people will test negative, of which 22022 are actually free of the condition.

Therefore the odds that a person who tests negative is actually free of the condition is 22022/22102, or about 99.6 in a hundred