1) When sending data over the internet there is a certain probability that a message will be corrupted. One way to improve the reliability of getting messages through is to use a Hamming Code. This involves sending extra data that can be used to check the main message. For example a 7 bit Hamming Code contains 4 bits of message data and 3 check bits. If only one of the bits is in error at the receiving end then mathematical techniques can be used to determine which one it is and apply a correction. Assume that you have a network connection for which the probability that an individual bit will get through without error is 0.66. What is the increase in the probability that a 4 bit message will get through if a 7 bit Hamming code is used instead of just sending the 4 bits? (i.e what is P(7 bits with 0 or 1 error) - P(4 bits with no error)?
2) Q Computers has invented quantum computers. Each computer contains an exotic sub-atomic particle. Unfortunately this particle decays in the same manner as all radioactive particles. Therefore an average quantum computer only lasts for 22 months. The University has purchased one of these computers and Professor Squiggle wants to use it for 7 months. When he tries to book it he finds that it is already booked out for the first 8 months. So he books it for the next 7 months. What is the probability that the computer will fail during the time that professor Squiggle is using it (not before and not after)?
This solution has step-by-step calculations with explanations to the statistics problems mentioned. All formulas and workings shown.