A Measurement of Randomness in Coin Tossing

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1 PHYS 0040 Brown University Spring 2011 Department of Physics Abstract A Measurement of Randomness in Coin Tossing A single penny was flipped by hand to obtain 100 readings and the results analyzed to check for bias or non-random patterns. Forty-seven flips resulted in heads. The longest repeating string in the data is six consecutive tails. the result of a given flip is not correlated with results one, two or three flips earlier. The results are consistent with coin tossing being an unbiased random process. Introduction Many unpredictable events are referred to as random when they are better described as complicated or poorly understood. Tossing a coin is clearly such a process, calculable in principle but hard to predict in practice. Flipping a coin is often used as a deliberately random process in decision making. It was decided to check, by direct observation, if coin flipping mimics the essential features of a random process. These features are a lack of bias and the lack of correlation of future outcomes with previous results [1]. The experiment, results and analysis are described below. Experiment A single 1999 United States penny was flipped by the author until 100 readings were obtained. the coin was flipped and caught with the left hand, inverted onto the back of the right hand and read. Dropped coins were read where they fell. Tosses during which the coin failed to rotate at least 180 about a horizontal axis were rejected as bas tosses. There were nine such rejected tosses. Strikingly, eight of these nine rejects are tails, hinting that some aspects of the process may be far from random. Of the 512 possible outcomes of nine sequential flips, only 20 patterns, or 4%, are as biased as the pattern of rejects. The results of the 100 successful flips are recorded in order in Table 1 as a string of H s and T s. Analysis Heads occur 47 times in the data, implying that the chance of obtaining heads on a single flip is P = 0.47 ± the 1σ uncertainty is calculated assuming Poisson statistics. This result is consistent with the process being unbiased, P = In Figure 1 the fraction of the first N flips which are heads is plotted against N (N = 1 to 100). The value lies near to or within 0.5 ± 1/ 2N, as expected for an unbiased random process. To test for unpredictability, an important feature of a random process, a search was made for correlations in the data. Assigning a value of X n = +1 or 1 respectively when the result of the n th flip is heads or tails, one can calculate the correlation function C m. C m = X n X n+m = m Σ100 m n=1 X n X n+m. (1) Clearly C 0 = 1, but one expects C m = 0 ± 2/N where m 0 for any random unbiased process [2]. The first few values of C m are listed in Table 2. All of the values are 1

2 consistent with zero. Since it is hard to believe that correlations could be present at large delay if they are not present right away, this analysis was not carried out to larger values of m. There are several long strings of consecutive heads or of tails in the data. To check if such a pattern is unexpected, the number of occurrences of strings of length j flips is plotted versus j in Figure 2 for visual comparison with a model. One can calculate the expected number of strings of heads or tails of a given length as follows. To flip j heads in a row, and no more than that, one must flip tails, then heads j times in a row, and then tails again. The chance of doing this in j = 2 flips is P j = 2 (j+2). The chance of getting a similar string of tails is identical, so the expected number of strings of length j in 100 flips is S j = (j+2) = 100. (2) 2j+1 This equation is plotted in Figure 2 along with the data. The two curves are in general agreement. Conclusions Casual repeated flipping of a coin seems to exhibit the most important features of a random process, in spite of mild evidence for correlations built into the process. The precision of our measurements is 7%, which is the expected accuracy for 100 flips. References [1] Data Reduction and Error Analysis for the Physical Sciences, P.R. Bevington and D. K. Robinson, McGraw Hill, [2] Everyday Random Processes, in PH156 Lab Manual, Brown University Department of Physics, 2001 (to be published). 2

3 Table 1. Results of Flipping a Coin 100 Times. THHTHHTHHHTHHHHTHTTHTHTTH HTTTHTTHHTTTTTTHHHTTTTTHH HTTTHTHHHHTTHHHTHTTTTTHHH TTTTHHTTHTHTTTHHHTHTTTHHT Failed Tosses (No rotation): TTTTTTHTT Table 2. Autocorrelation Coefficients C 1 = X n X n ± 0.15 C 2 = X n X n ± 0.15 C 3 = X n X n ± 0.15 Uncertainties calculated assuming a Poisson distribution. 3

4 Figure 1. The fraction of flips which are heads The cumulative fraction of flips which are heads is plotted as a function of the number of flips. The dot-dash straight line indicates the expected average value, one half. The two dashed curves show the range of values expected. Notice that the data lie near or within the limits. 4

5 Figure 2. Length distribution of consecutive strings The heavy line shows the number of strings of a given length of consecutive heads or consecutive tails plotted as a function of the string length. the dashed line shows the theoretical length distribution. The agreement is reasonable. 5