You are here:

Question
My question is how to calculate probability when confronted by experts having an argument about a particular subject (in cases where there are only 2 possible answers, such as yes/no questions).
For example: A, B and C are all expert doctors. When each of them (individually) gives a diagnosis (in a yes/no question), the chance of accuracy is 90%, or 9/10. In a case where A and B argue for a certain treatment, and C argues against, what is the chance that A and B are correct? Originally I worked out that the answer is 9/10, but when constructing the general formula I realised that if 11 doctors were arguing against 10 doctors the answer would be the same - 9/10 - which is counter-intuitive and wrong. I then realised that the more doctors who are at loggerheads over a question indicates that this question is more difficult than the average one, in which case the chance of accuracy of all doctors goes down. (90% is only the success rate for the average question, not for the more difficult ones.)

My question is: given these facts, is there any way to give a formula which will tell me the probability of accuracy given the number of doctors on both sides of the issue (assuming that all doctors have the same expertise, that they answer 90% of questions correctly)?

If anyone could help me I would appreciate it!

Given that the chance of being right is always 0.9,
the chance of being wrong is 1 - 0.9 = 0.1.

It would follow a binomial distribution with N doctors.
The chance of n doctors being right would be (N choose n)(0.9^n)[0.1^(N-n)].

If the chances were desired for at least n=a doctors being right,
the sum of the probabilities n=a, n=a+1, n=a+2, ... n=n.

Volunteer

#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.