Heads and Runs in Multiple Tossing of a Coin and Contingency Table, Statistical Note 40

List the possible outcomes in three tosses of a coin, categorize them by the number of number of heads and the number of runs in the sequence and prepare a contingency table.

Tossing of a coin thrice has two to the power three equal to eight possible outcomes (refer to my statistical note 9). Every outcome is measured using two discrete random variables – the number of heads and the number of runs. The outcomes have number of heads ranging from zero to three as mutually exclusive categories of response as shown in Table 1. A run is a sequence of flips of the same face of a coin. The number of runs ranges from one to three as mutually exclusive categories of response (Table 1). Example, an outcome HTH has three runs, as every toss has a different face than the previous toss. A contingency table, also known as the cross tabulation, crosstab or two-way table counts the number observations for each category of two variables.

Table 1: Possible outcomes in three tosses of a coin by number of heads and number of runs

The contingency table presents the number of possible outcomes by number of heads and number of runs (Table 2).

Central values of both variables have high probability of occurrence. Example, outcomes with one and two heads respectively have higher probability of occurrence, three by eight. Likewise, outcomes with two runs have higher probability of occurrence, four by eight. Refer to my statistical note 3 for probability and contingency table.