Practise defining trials, outcomes, events and sample spaces, calculating probabilities or expected frequencies, and checking validity in context.
This question is about applying ideas from logarithms, calculus and probability to an unfamiliar mathematical theory called information theory.
Claude Shannon developed a mathematical theory called information theory to measure the information gained when random events occur. He defined the information, I, that is gained when an event with probability p occurs as
where 0<p≤1. For example, no information is gained ( I=0 ) when an event is certain to occur(p=1).
A computer selects at random an integer x from 1 to 10, inclusive. Each outcome is equally likely.
Alessia is trying to determine the value of x and asks if x is odd.
Write down the probability that x is odd.
105(=21)
Alessia is told that x is odd. Find how much information Alessia gains.
The computer then selects at random an integer y from 1 to 10 , inclusive. Each outcome is equally likely.
Daniel is trying to determine the value of y and asks if y is 7 . He is told that it is not 7 .
attempt to substitute p= their (b)(i) into I=−lnp=0.693(0.693147…,−ln(21))
Find how much information Daniel gains.
If a random variable has n possible outcomes with probabilities p1,p2…pn, then the expected information gained, E(I), is defined as
109=0.105(0.105360…,−ln(109))