Three Coins
Here, I added yet another coin, flipped these three coins 100 times, and recorded the outcomes like the previous experiment. Microsoft Excel randomly chose 0 or 1 100 times for each of the three coins, where 0 is heads and 1 is tails. Once again, I took the sum of the three coins. The possible sums were as follows: 0 + 0 + 0 = 0 (all three coins were heads), 0 + 0 + 1(in no particular order) = 1 (two coins landed on heads and the other on tails), 0 + 1 + 1(in no particular order) = 2 (one coin landed on heads and the other two on tails), or 1 + 1 + 1 = 3 (all three coins landed on tails). Next, I took the frequency of the possible sums, and the spreadsheet clearly shows the outcomes of this (remembering that I have only shown the first few flips). Heads appeared on all three coins 12 out of 100 times (12%), heads appeared on two of the three coins 42 out of 100 times (42%), heads appeared on one of the three coins 38 out of 100 times (38%), and heads did not appear on any of the three coins 8 out of 100 times (8%).
Probability of Getting 0 Heads:
Empirical: .08
Theoretical: .125
Probability of Getting 1 Head:
Empirical: .38
Theoretical: .375
Probability of Getting 2 Heads:
Empirical: .42
Theoretical: .375
Probability of Getting 3 Heads:
Empirical: .12
Theoretical: .125
Let's look at another simulation of the same experiment:
Probability of Getting 0 Heads:
Empirical: .11
Theoretical: .125
Probability of Getting 1 Head:
Empirical: .33
Theoretical: .375
Probability of Getting 2 Heads:
Empirical: .42
Theoretical: .375
Probability of Getting 3 Heads:
Empirical: .14
Theoretical: .125