Author: Awais Ahmad (


Question: Prove with experiment that the Probability of a fair coin is 0.5 or 50%.


Step 1: Take a fair coin and toss it 200 times and record each observation and the outcomes thereof. The following observations were recorded by tossing a fair coin 200 times:

For the time being, for our convenience, we have classified the observations of the experiment with a difference of 25.


Head of the Fair Coin = N

Tail of the Fair Coin = M

As we see as per Table, First column shows the number of observations taken from the experiment, classified with the difference of 25. Second column shows the appearance of Head (N) or Tail (M) during the experiment.

Step 2: Now we perform calculations of Probability in further details by the table given below:

Colum 1 of Table shows the Number of Experiments’ Lower and Upper Limits (The data of 200 observations is classified with a difference of 5 for detailed analysis). Second column shows outcomes of appearing Head (N) or Tail (M), and Cumulative Outcomes. TOTAL Colum shows the number of experiments during a particular time. Next column shows the probabilities of appearing N or M of the coin and their Cumulative Probabilities. These probabilities are calculated by the following formula:

P (N) = N / (TOTAL)

P (M) = M / (TOTAL)

Step 3: Then P (N) and P (M) showed the cumulative probabilities of N and M respectively. The point should be noted that, before experiment performed, the Probabilities of both N and M were Zero. Last column shows the number of Cumulative Observations/Outcomes of the experiment. Last Row TOTAL showed that the experiment was repeated for 200 times, among which 98 times, N appeared and M appeared for 102 times.

Now we plot these values on the line graph, and can show the probabilities of N, M and both N & M on the same plot area.


From the experiment Data and Graphical Representations, we can see that the Probabilities of Head (N) and Tail (M) are about equal to 0.5 (50%) when we repeat the same experiment for 200 times. So it is proved that the Probability of a fair coin is 0.5 (50%).

Note: If the same experiment is provided for more than 200 times, we can get more accurate results.


Question from Statistics for Business and Economics – by David R. Anderson (Author), Dennis J. Sweeney (Author), Thomas A. Williams (Author), Jeffrey D. Camm (Author), James James J. Cochran (Author)

Experiment on Fair Coin

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