Chapter 1, Problem 1DHGP

Use a random-number table or random-number generator to simulate tossing a fair coin 10 times. Generate 20 such simulations of 10 coin tosses. Compare the simulations. Are there any strings of 10 heads? Of 4 heads? Does it seem that in most of the simulations, half the outcomes are heads? Half are tails? In Chapter 6, we will study the probabilities of getting from 0 to 10 heads in such a simulation.

To determine

The random number generator to simulate 20 samples of 10 coins tosses and explain 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

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T

H

H

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H

H

T

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H

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H

H

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T

H

H

T

H

H

H

H

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T

H

H

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T

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H

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H

H

It does not seem that any string has 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

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H

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T

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H

H

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T

H

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H

H

H

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H

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H

H

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H

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H

H

H

H

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H

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H

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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 seem that in most of simulations, half the outcomes are head or tails.