Question

Short answer

The probability of getting three heads and three tails is (a) \(\frac{5}{{16}}\) , and the probability of getting six heads is (b) \(\frac{1}{{64}}\).

Step-by-step solution

Understanding microstates

A microstate is an arrangement between several particles at a single instant.

Constructing the table showing the number of microstates and macrostate  

The total number of possible microstates after tossing the coin six times on the floor is \({26} = 64\).

The following table shows the microstates and macrostates.

Macrostate	Possible Microstate-Head (H) and Tail (T)						Microstates
6 head 0 tails	HHHHHH						1
5 head 1 tails	HHHHHT	HHHHTH	HHHTHH	HHTHHH	HTHHHH	THHHHH	6
4 head 2 tails	HHHHTT	HHHTHT	HHTHHT	HTHHHT	THHHHT		15
	HHHTTH	HHTHTH	HTHHTH	THHHTH	HHTTHH
	HTHTHH	THHTHH	HTTHHH	THTHHH	TTHHHH
3 head 3 tails	HHHTTT	HHTHTT	HTHHTT	THHHTT	HHTTHT		20
	HTHTHT	THHTHT	HTTHHT	THTHHT	TTHHHT
	TTTHTH	TTHTHH	THTTHH	HTTTHH	TTHHTH
	THTHTH	HTTHTH	THHTTH	HTHTTH	HHTTTH
2 head 4 tails	TTTTHH	TTTHTH	TTHTTH	THTTTH	HTTTTH		15
	TTTHHT	TTHTHT	THTTHT	HTTTHT	TTHHTT
	THTHTT	HTTHTT	THHTTT	HTHTTT	HHTTTT
1 head 5 tails	TTTTTH	TTTTHT	TTTHTT	TTHTTT	THTTTT	HTTTTT	6
0 head 6 tails	TTTTTT						1

Evaluation of probability of obtaining three heads and three tails  

The probability of getting three heads and three tails is calculated below:

\(\begin{array}{l}P = \frac{{20}}{{64}}\\P = \frac{5}{{16}}\end{array}\)

Thus, \(\frac{5}{{16}}\) is the required probability.

Evaluation of probability of obtaining six heads  

The probability of getting six heads is calculated below:

\(P = \frac{1}{{64}}\)

Thus, \(\frac{1}{{64}}\) is the required probability.