Question

An examiner sets twelve problems, and tells the class that the exam will consist of six of them, selected at random. Gavin memorises the solutions to eight problems in the list, but cannot solve any of the others. What is the chance he gets four or more correct?

Answer 1

The chances Gavis get four or more correct problems is 8/11 or 72.72%

Explanation:

The exam is composed of 6 problems out of 12 possible cases (Pc=12). There are 2 groups of problems:

The 8 problems that Gavin has the answer (Problems A).

The 4 problems that Gavin hasn´t the answer (problems B).

Therefore:

P(A≥4)= P(A=4) ∪ P(A=5) ∪ P(A=6) = P(A=4) + P(A=5) + P(A=6)

Before we start analyzing the problem, we have to understand that problems in the exam are selected at random, but a problem can´t be selected twice. therefore picking a specific problem will reduce the pool of that specific group and of the total number of available problems.

If we call
`X_(ij)` to the probability of an answer of the X group to be the i° picked problem from the j° picked problem of that group:

`X_(ij)=(N_(GRX)-j+1)/(12-i+1)` with
`N_(GRX)` the total number of problems in that group.

We analyze now 3 different problems:

`P(A=6)=A_(11) \cdot A_(22) \cdot A_(33) \cdot A_(44) \cdot A_(55) \cdot A_(66)\\P(A=6)=(8)/(12) (7)/(11) (6)/(10) (5)/(9) (4)/(8) (3)/(7) = (1)/(33)`

`P(A=5)=A_(11) \cdot A_(22) \cdot A_(33) \cdot A_(44) \cdot A_(55) \cdot B_(61)+ A_(11) \cdot A_(22) \cdot A_(33) \cdot A_(44) \cdot B_(51) \cdot A_(65)+...+B_(11) \cdot A_(21) \cdot A_(32) \cdot A_(43) \cdot A_(54) \cdot A_(65)\\`

`P(A=5)=(8)/(12) (7)/(11) (6)/(10) (5)/(9) (4)/(8) (4)/(7) + (8)/(12) (7)/(11) (6)/(10) (5)/(9) (4)/(8) (4)/(7) + ... + (4)/(12) (8)/(11) (7)/(10) (6)/(9) (5)/(8) (4)/(7) = 6 (8 \cdot 7 \cdot 6 \cdot 5 \cdot4 \cdot 4)/(12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7)=(8)/(33)`

For P(A=4) we can take the solution from P(A=5) and say that:

`P(A=4)=c \cdot (4 \cdot 3 \cdot 8 \cdot 7 \cdot 6 \cdot5 )/(12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7)` where "c" is the combinatorial of 2 problems B with 4 problems A. In this case "c" is 15, therefore:

`P(A=4)=(8)/(11)`

`P(A \ge4)= P(A=4) + P(A=5) + P(A=6) =(1)/(33) +(8)/(33) +(5)/(11) = (8)/(11)`