In a criminal investigation (hypothetical) the question is: "Was this home video made using this camcorder?"
The camcorder has a pickup device (charged coupled device or CCD) consisting of a rectangular array of 300,000 sensors or pixels. Four of these are "hot pixels." In every frame shot with this camcorder these four hot pixels although difficult see will always appear in the same locations.
If every one of the total of 300,000 pixels has an equal chance of being one of the four hot pixels, the odds that out of a total of just four hot pixels they would be at these four specific locations in the array is 1 in 300,000 to the 4th power = 8.1 x 10 to the 20th
1) For a particular high-paying customer, Quality Control at the factory that makes the sensor arrays will "Pass" such a 300,000 pixel array if it has no more than 4 isolated hot pixels. However, adjacent hot pixels are much more readily seen, and therefore any arrays with even two adjacent (either side-by-side or one above the other) hot pixels will be rejected.
2) For less discriminating customers (and paying less), Quality Control will "Pass" those sensor arrays if two hot pixels are adjacent and the total of hot pixels does not exceed 4, but it will reject any arrays having three adjacent hot pixels (either all in line or two side-by-side and the third either above or below one of the first two).
3) For even less discriminating customers Quality Control will "Pass" such a sensor array even if it has three adjacent hot pixels and no more than a total of 4 hot pixels. However, the array will be rejected if all four hot pixels are touching (4 possible configurations).
In these three situations the probability of 1 in 8.1 x 10 to the 20th is too high. Because of the limitations made by Quality Control, not all possible locations of 4 hot pixels in the 300,000 pixel array are permitted. In situation 1), one must subtract from that figure the total number of possible combinations of two adjacent pixels (either side-by-side or one above the other) in an array of 300,000 pixels. For situation 2), one must subtract the total number of possible combinations of three adjacent pixels (both all in line or two side-by-side and the third either above or below one of the first two). For situation 3) one must subtract the total number of possible combinations where all 4 hot pixels are in contact (4 possible configurations).
Show how you would calculate the numbers in these three situations and explain your reasoning.