Homework SoursePart 1 Coin tosses Consider a set of N coins If we toss each
Part 1: Coin tosses
Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates) that the N coins can come down is 2 x 2 x ... (N times)=2^N. While this is interesting, this is not the number we want. Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that came up with that particular number of heads and tails?
A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates:
-4H, 0T
-3H, 1T
-2H, 2T
-1H, 3T
-0H, 4T
Part 1: Coin tosses
Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates) that the N coins can come down is 2 x 2 x ... (N times)=2^N. While this is interesting, this is not the number we want. Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that came up with that particular number of heads and tails?
A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates:
-4H, 0T
-3H, 1T
-2H, 2T
-1H, 3T
-0H, 4T
Consider a set of N coins. If we toss each coin, each has two ways of coming down, H or T. Since the first coin can come down 2 ways, and the second coin can come down 2 ways, etc., the number of different ways (microstates) that the N coins can come down is 2 x 2 x ... (N times)=2^N. While this is interesting, this is not the number we want. Rather, we want to know if we choose a particular macrostate (a given number of heads and tail) how many microstates correspond to that macrostate. That is, how many different ways could you get a string of coin flips that came up with that particular number of heads and tails?
A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates:
-4H, 0T
-3H, 1T
-2H, 2T
-1H, 3T
-0H, 4T
Solution
-4H, 0T - (HHHH) -- 1 way
-3H, 1T -(HHHT)(THHH)(HTHH)(HHTH)- 4 ways
-2H, 2T - (HHTT)( TTHH)(HTTH)(THHT)(THTH) (HTHT- 6 ways
-1H, 3T - (HTTT) (THTT) (TTHT)(TTTH)-4
-0H, 4T (TTTT) - 1 way
Verify to check the total no of ways = 16 = 24
