What is the probability of getting correct directions at MIT?

In summary: Therefore, the probability that the answer is correct is 6/12 or 1/2. In summary, the probability that the passer-by's answer of East being the exit from campus is correct is 1/2.
  • #1
Eidos
108
1

Homework Statement



You are lost in the campus of MIT, where the population is entirely composed of brilliant students and absent-minded professors. The students comprise two-thirds of the population,
and anyone student gives a correct answer to a request for directions with probability [tex]\frac{3}{4}[/tex] (Assume answers to repeated questions are independent, even if the question and the person asked are the same.) If you ask a professor for directions, the answer is always false.

You ask a passer-by whether the exit from campus is East or West. The answer is East. What is the probability this is correct?

Homework Equations



[tex]P(A|B)=\frac{P(B|A)P(A)}{P(B)}[/tex] {Baye's Theorem}

[tex]P(A\cap B) = P(A)P(B)[/tex] {For independent events A and B}

[tex]P(A\cup B) = P(A)+P(B)-P(A\cap B)[/tex]

The Attempt at a Solution


My approach was to use Baye's Theorem. The problem is that I don't have any prior probabilities.

Let P(P) be the probability that the person is a prof.
P(S) '' '' is a student.
P(E) '' '' answer is East.
P(T) " " correct answer is given.

Now

[tex]P(T|E) = \frac{P(E|T)P(T)}{P(E)}[/tex]

with

[tex]P(T)=P(S \cap T \cup P \cap T)[/tex]

and [tex]P(E) = 0.5[/tex].

Is this the right way to go about it?

As a heuristic, a later question says that if you ask the same person again, and they answer East, you need to show that the probability that East is True is 1/2.

Any points in the right direction would be most welcome :smile:
 
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  • #2
Using a tree might make this easy to solve.

P(student) * P(correct | student) = 2/3*3/4 = 6/12

P(student) * P(~correct | student) = 2/3*1/4 = 2/12

P(prof) * P(correct | prof) = 1/3*0 = 0

P(prof) * P(~correct | prof) = 1/3*1 = 4/12

Only one branch offers a correct answer since professors always answer incorrectly.
 

1. What is "Lost at MIT probability"?

"Lost at MIT probability" refers to a mathematical concept used to calculate the likelihood of a person becoming lost while navigating the campus of MIT (Massachusetts Institute of Technology). It takes into account factors such as the size and complexity of the campus, as well as the person's familiarity with the area.

2. How is "Lost at MIT probability" calculated?

The formula for calculating "Lost at MIT probability" involves multiplying the probability of getting lost at any given point on the campus by the total number of points a person must navigate through. This gives an estimate of the overall likelihood of getting lost while navigating the campus.

3. Is "Lost at MIT probability" a real concept or just a hypothetical scenario?

"Lost at MIT probability" is a real concept used by researchers and mathematicians to study the probability of getting lost in complex environments. It can also be applied to other situations, such as getting lost in a city or a maze.

4. Can "Lost at MIT probability" be used to prevent people from getting lost at MIT?

While "Lost at MIT probability" can give an estimate of the likelihood of getting lost, it is not a tool for preventing people from getting lost. However, it can be used by campus planners and designers to create more navigable and user-friendly environments.

5. Are there any other factors that can affect "Lost at MIT probability"?

Yes, in addition to the size and complexity of the campus, factors such as weather conditions, time of day, and the individual's physical and mental state can also impact the likelihood of getting lost. These variables can be incorporated into the calculation to provide a more accurate estimate.

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