Chapter 1, Problem 1DHGP

To determine

The random number generator to simulate 20 samples of 10 coins tosses and explains the comparison of simulations.

Answer to Problem 1DHGP

Solution: The 20 simulation of 10 coin tosses is obtained as:

$s_1$

$s_2$

$s_3$

$s_4$

$s_5$

$s_6$

$s_7$

$s_8$

$s_9$

$s_{10}$

$s_{11}$

$s_{12}$

$s_{13}$

$s_{14}$

$s_{15}$

$s_{16}$

$s_{17}$

$s_{18}$

$s_{19}$

$s_{20}$

H

H

T

T

T

H

H

H

H

H

T

T

H

H

T

T

H

T

H

T

H

H

T

H

H

T

H

H

T

T

T

T

T

T

T

T

H

H

T

T

T

H

H

T

T

H

T

T

H

T

H

T

T

T

H

H

T

H

H

T

H

H

H

T

T

T

H

T

T

T

T

T

H

T

H

T

T

T

T

T

H

H

T

T

T

H

H

T

H

H

T

H

H

H

T

T

T

H

H

T

H

H

H

H

H

T

H

H

T

T

T

H

H

T

T

H

T

T

H

T

H

T

H

T

H

H

T

T

T

H

H

T

H

T

H

H

T

T

T

T

T

H

H

T

H

H

T

T

H

H

H

H

T

T

H

H

T

H

H

H

H

T

H

H

H

H

H

H

T

H

H

T

T

T

T

H

T

T

T

H

T

T

T

T

T

T

T

T

H

H

T

T

H

H

T

T

H

H

H

H

It does not seem that any string of 10 heads. There are 4 strings of four heads. Only 4 strings have half heads and half tails.

Explanation of Solution

There are several ways to assign numbers to the two outcomes because coin is fair. Suppose, quarter is fair ‘0’ digit is assigned to the outcome heads (H) and ‘1’ digit to the outcome tails (T).

To simulate the outcomes of tossing a coin 10 times by using Excel as follows:

Select the cell that to use to generate the random number between 0 and 1 in Excel. Select cell A1.

Enter the formula =RANDBETWEEN(0,1) in cell A1.

This formula is then copied down from A1 to A10.

Copied these 10 cells in column B to T to obtain 20 simulations.

The 20 simulation of 10 coin tosses is obtained as:

$s_1$	$s_2$	$s_3$	$s_4$	$s_5$	$s_6$	$s_7$	$s_8$	$s_9$	$s_{10}$	$s_{11}$	$s_{12}$	$s_{13}$	$s_{14}$	$s_{15}$	$s_{16}$	$s_{17}$	$s_{18}$	$s_{19}$	$s_{20}$
0	0	1	1	1	0	0	0	0	0	1	1	0	0	1	1	0	1	0	1
0	0	1	0	0	1	0	0	1	1	1	1	1	1	1	1	0	0	1	1
1	0	0	1	1	0	1	1	0	1	0	1	1	1	0	0	1	0	0	1
0	0	0	1	1	1	0	1	1	1	1	1	0	1	0	1	1	1	1	1
0	0	1	1	1	0	0	1	0	0	1	0	0	0	1	1	1	0	0	1
0	0	0	0	0	1	0	0	1	1	1	0	0	1	1	0	1	1	0	1
0	1	0	1	0	0	1	1	1	0	0	1	0	1	0	0	1	1	1	1
1	0	0	1	0	0	1	1	0	0	0	0	1	1	0	0	1	0	0	0
0	1	0	0	0	0	0	0	1	0	0	1	1	1	1	0	1	1	1	0
1	1	1	1	1	1	1	1	0	0	1	1	0	0	1	1	0	0	0	0

The outcomes (H and T) associated with 20 samples as:

$s_1$

$s_2$

$s_3$

$s_4$

$s_5$

$s_6$

$s_7$

$s_8$

$s_9$

$s_{10}$

$s_{11}$

$s_{12}$

$s_{13}$

$s_{14}$

$s_{15}$

$s_{16}$

$s_{17}$

$s_{18}$

$s_{19}$

$s_{20}$

H

H

T

T

T

H

H

H

H

H

T

T

H

H

T

T

H

T

H

T

H

H

T

H

H

T

H

H

T

T

T

T

T

T

T

T

H

H

T

T

T

H

H

T

T

H

T

T

H

T

H

T

T

T

H

H

T

H

H

T

H

H

H

T

T

T

H

T

T

T

T

T

H

T

H

T

T

T

T

T

H

H

T

T

T

H

H

T

H

H

T

H

H

H

T

T

T

H

H

T

H

H

H

H

H

T

H

H

T

T

T

H

H

T

T

H

T

T

H

T

H

T

H

T

H

H

T

T

T

H

H

T

H

T

H

H

T

T

T

T

T

H

H

T

H

H

T

T

H

H

H

H

T

T

H

H

T

H

H

H

H

T

H

H

H

H

H

H

T

H

H

T

T

T

T

H

T

T

T

H

T

T

T

T

T

T

T

T

H

H

T

T

H

H

T

T

H

H

H

H

The above random sample is obtained with replacement. There is no any string (sample) of 10 heads. There are 4 samples of 4 heads. These four samples are $s_8,s_{11},s_{15}\text{ and }{s}_{19}.$

There are 4 samples of 5 heads and 5 tails. These four samples are $s_5,s_9,s_{16}\text{ and }{s}_{18}.$ It does not seems that in most of simulations, half the outcomes are head or tails.