Dentist's agree

[BMAD]

Ten out of 17 people suffer from plaque buildup, but Four out of five dentist agree that the new BMAD formula for mouth rinse will cure 95% of the people with gingivitis of plaque build up and is 85% effective for those who do not have gingivitis. If a random person is selected what is the probability they would be effectively treated of plaque buildup?

[bonanova]

I'll make these assumptions.

1. "Effectively treated" means "cured."
   Both terms are used in the OP so I take them to mean the same thing.
   
2. Those without plaque cannot be cured of plaque.
   So, off the top, 7 out of 17 people will not be "effectively treated."
   
3. "Four out of five dentists agree ... x." means "x has a probability of 0.8."

If we were to treat a representative group of 17 people,

Five would have plaque and gingivitis. They would contribute 5/17 x .8 x .95 effectively treated cases.

Five would have plaque but no gingivitis. They would contibute 5/17 x .8 x .85 effectively treated cases.

Seven would have no plaque. They would contrubute 7/17 x 0 effectively treated cases.

Effective treatment probability of a random person is therefore

p(effective) = 5/17 x .8 x .95 + 5/17 x .8 x .85 + 7/17 x 0 = 7.2/17 = 0.4235.

[bonanova]

Do we know the prior probability of having gingivitis?

[BMAD]

Oh. assume 50 50 chance

[m00li]

I think it also depends on what the remaining fifth of dentists think too. e.g. they might think that the BMAD formula is effective on 100% of plaque buildups (case A extreme) or totally ineffective (case B extreme) whether a person has or doesn't have plaque buildup.

case A extreme:

probability that the person has plaque = P(has plaque) = 10/17

P(effectively treated) =

P(BMAD's formula behaves as 4/5 dentists predict)*[P(has gingivitis and is effectively treated) +P(doesnt have gingivitis and is eff. treated)] +

P(BMAD's formula behaves as 1/5 dentists predict)*[P(has gingivitis and is effectively treated) +P(doesnt have gingivitis and is eff. treated)] +

P(BMAD's formula doesnt behave as any of the dentist's predict)*[P(has gingivitis and is effectively treated) +P(doesnt have gingivitis and is eff. treated)]

Another Assumption (the last summand in the above sum is 0)

therefore, P(eff. treated)= 0.8(0.5*0.95 + 0.5*0.85) + 0.2(1) = 0.4(1.8) + 0.2 = 92%

case B extreme:

Under similar assumptions

P(eff. treated)= 0.8(0.5*0.95 + 0.5*0.85) + 0.2(0) = 0.4(1.8) + 0.2 = 72%