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Chapter 15: Q50P (page 412)

(II) Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining

(a) three heads and three tails, and

(b) six heads?

Short Answer

Expert verified

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

01

Understanding microstates

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

02

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

03

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.

04

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.

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