Consider n=23 independent flips of a fair coin. say that a changeover occurs whenever an outcome differs from the one preceding it. for example, if n=6 and the outcome is t h t t h t, then there is a total of 4 changeovers. find the expected number of changeovers for n=23.

Adrawer contains 12 identical white socks, 18 identical black socks and 14 identical brown socks. what is the least number of socks you must choose, without looking, to be certain that you have chosen two socks of the same colour?

Francisco's credit card has an apr of 28.98%, calculated on the previous monthly balance. his credit card record for the last 7 months is shown in the table below francisco is trying to figure out what this all means. answer the following questions: what is the total amount that francisco has paid in interest over the 7 months? what is the total amount that francisco has made in payments over the 7 months? how much of francisco's payments have gone to paying down the principal on his account? what is francisco's new balance at the end of month 8? what will francisco be charged for interest for month 8?

An expression for the average of x 2x and 3y