Question

Consider the random experiment of tossing a coin once. There are two possible outcomes for this experiment, namely a head (H) or a tail (T).

a. Repeat the random experiment five times- that is toss a coin five times- and record the information required in the following table. (The third and fourth columns are for running totals and running proportions, respectively)

b. Based on your five tosses, what estimate would you give for the probability of a head when this coin is tossed once? Explain your answer.

c. Now toss the coin five more times and continue recording in the table so that you now have entries for tosses \(1-10\). Based on your \(10\) tosses, what estimate would you give for the probability of a head when this coin is tossed once? Explain your answer.

d. Now toss the coin \(10\) more times and continue recording in the table so that you now have entries for tosses \(1-20\). Based on your \(20\) tosses, what estimate you would give for the probability of a head when this coin is tossed once? Explain your reasoning.

e. In view of your results in parts (b)-(d), explain why the frequentist interpretation cannot be used as the definitions of probability.

Short answer

Part a. \(\frac{4}{5}=0.8\)

Part b. \(P(H)=0.8\)

Part c. \(P(H)=0.6\)

Part d. \(P(H)=0.55\)

Part e. The frequent interpretation cannot be used as the definition of probability because probability changes as number of tosses changes.

Step-by-step solution

Part a. Step 1. Given information

\(\frac{4}{5}\)

Part a. Step 2. Calculation

Toss a coin five times and record the information as shown in following table:

Toss	Outcome	Number of heads	Proportion of heads
\(1\)	H	\(4\)	\(\frac{4}{5}=0.8\)
\(2\)	H
\(3\)	H
\(4\)	T

\(5\)

H

Part b. Step 1. Given information

\(\frac{4}{5}=0.8\)

Part b. Step 2. Calculation

When you tossed a coin \(5\) times, you got \(4\) heads or in \(N=5\) tosses, \(f=4\) were heads. Therefore, based on five tosses, the probability of a head when this coin is tossed once is:

\(P(H)=\frac{f}{N}\)

\(=\frac{4}{5}\)

\(=0.8\)

Part c. Step 1. Given information

\(N=10\)

Part c. Step 2. Calculation

Toss a coin five more times and record the information as shown in following table:

Toss	Outcome
\(1\)	H
\(2\)	H
\(3\)	H
\(4\)	T
\(5\)	H
\(6\)	T
\(7\)	T
\(8\)	H
\(9\)	T

\(10\)

H

When you tossed a coin \(10\) times, you got \(6\) heads or in \(N=10\) tosses, \(f=6\) were heads. Therefore, based on ten tosses, the probability of a head when this coin is tossed once is:

\(P(H)=\frac{f}{N}\)

\(=\frac{6}{10}\)

\(=0.6\)

Part d. Step 1. Given information

\(N=20\)

Part d. Step 2. Calculation

Toss a coin ten more times and record the information as shown in following table:

Toss	Outcome	Toss	Outcome
\(1\)	H	\(11\)	H
\(2\)	H	\(12\)	H
\(3\)	H	\(13\)	T
\(4\)	T	\(14\)	T
\(5\)	H	\(15\)	H
\(6\)	T	\(16\)	T
\(7\)	T	\(17\)	T
\(8\)	H	\(18\)	T
\(9\)	T	\(19\)	H

\(10\)

H

\(20\)

H

When you tossed a coin \(20\) times, you got \(11\) heads or in \(N=20\) tosses, \(f=11\) were heads. Therefore, based on ten tosses, the probability of a head when this coin is tossed once is:

\(P(H)=\frac{f}{N}\)

\(=\frac{11}{20}\)

\(=0.55\)

Part e. Step 1. Given information

\(N=10, N=20\)

Part e. Step 2. Calculation

The frequent interpretation cannot be used as the definition of probability because probability changes as number of tosses changes.