Example of Event Independence in Discrete Mathematics
Explore an example in discrete mathematics where events E and F are examined for independence based on randomly generated bit strings. Understand the calculation process and conclusion drawn regarding the independence of these events.
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Discrete Math: Example 1 of Independence
Example 1 Suppose E is the event that a randomly generated bit string of length four begins with a 1 and F is the event that this bit string contains an even number of 1s. Are E and F independent, if the 16 bit strings of length four are equally likely?
Solution There are eight bit strings of length four that begin with a one: 1000, 1001, 1010, 1011, 1100, 1101, 1110, and 1111. There are also eight bit strings of length four that contain an even number of ones: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111. Because there are 16 bit strings of length four, it follows that p(E) = p(F) = 8/16 = 1/2. Because E F = {1111, 1100, 1010, 1001}, we see that p(E F ) = 4/16 = 1/4. Because p(E F ) = 1/4 = (1/2)(1/2) = p(E)p(F ), we conclude that E and F are independent.
References Discrete Mathematics and Its Applications, McGraw-Hill; 7th edition (June 26, 2006). Kenneth Rosen Discrete Mathematics An Open Introduction, 2nd edition. Oscar Levin A Short Course in Discrete Mathematics, 01 Dec 2004, Edward Bender & S. Gill Williamson
