As seen in Problem D1, a random player has a 50% chance of hitting a line with a push. Consequently, this problem can use the binomial probability model; making four random pushes is equivalent to tossing a fair coin four times. Note 10
Here is the tree diagram for four pushes (H represents a hit, and M represents a miss):
Here is the probability table for four pushes:
Note that we've added columns to indicate decimal values and percentages for the mathematical probabilities, which are our expectations for the experimental probability.
How do the results from our player compare with the mathematical probabilities for our random player? Once again, here are the experimental results from 100 rounds of Push Penny:
To compare these results with the random player's, you'll need to summarize them in a probability table and count the frequency of scores 0, 1, 2, 3, and 4. To make it easier to compare these frequencies with the random player's probabilities, let's calculate them as decimal proportions: Note 11
Too bad for our competitor -- there are only very slight differences between the proportions in the experimental data and the probabilities for a random player. Therefore, we do not have strong evidence that our competitor has developed any skill in playing the game. Indeed, our competitor's skills do not appear to be demonstrably greater (or weaker) than the random player's. (We'll break the news gently.)