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**In This Pa****rt: Developing the Mathematical Probability
**Let’s return to the statistics question we considered in Part A: Can a person develop skill at playing a game of chance like Push Penny? See Note 9 below.

Let’s say that a very determined person played 100 rounds of Push Penny with the following results (the number of hits out of four pushes per round):

Note: This is an ordered list (i.e., the first round was three hits out of four pushes, the second was two out of four, and so on).

Do these results suggest that our player has developed skill in playing the game? How might we analyze these data to answer this question? One approach would be to compare these 100 scores to the scores for a player who has no skill — in other words, a player who is just making “random” pushes.

In order to do this, you must have a description of a random player — specifically, you need to know the probability that a random push will hit a line.

**Problem D1
**What do you think the probability is that a random push will hit a line? Remember that each line is exactly two coin diameters apart. It may be helpful to experiment with a quarter on your Push Penny board and to examine this illustration. What percentage of the total area of the board is shaded?

**In This Part: Experimental Probability**

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. See Note 10 below.

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: See Note 11 below.

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.)

**Problem D2
**Here is the summary of scores of 100 rounds of another player’s attempt to master Push Penny. Do these scores suggest that this player has developed some serious Push Penny skill?

**Note 9**

In Part D, we return to the statistics question in Part A, based on the game of Push Penny: “After several practices of Push Penny, have you developed skill in playing the game?”

One approach is to compare the data (the 100 scores of a player) to the expected scores of a “random” player (one with no particular skill, who is making “random” pushes). This strategy requires a model for a “random” player, which must be based on probabilities, because there is randomness in the outcomes of the games.

A game consists of four pushes. First, you consider the probability that a single random push will hit a line. Experiment with a quarter on the Push Penny board to investigate this. The key is to discover that the lines are uniformly spaced (the distance between lines is equal to two times the diameter of a quarter). By moving a quarter perpendicularly to the lines, you’ll discover that the coin is touching a line half of the time and not touching a line half of the time.

**Note 10**

The leap from playing the game to describing the outcomes with a binomial model can be challenging. To further illustrate how the coin-tossing model describes a random player, the tree diagram is revisited. When the tree diagram for possible scores of this game is discovered to be the same as the tree diagram for the possible number of heads on four coin tosses, the equivalence of the modes may then be more clear.

**Note 11**

This analysis is based on investigating how the experimental results compare with the mathematical probabilities. As you’ll see, there are very slight differences between the player’s scores and the probabilities for a random player. Therefore, there does not appear to be strong evidence that our competitor is any better (or worse) than a random player.

This type of analysis is referred to as “goodness of fit” because we are asking how well the model fits the data. A more advanced analysis would require a Chi-Square test, which considers whether the observed differences between the experimental proportions and the theoretical probabilities can be explained by the random variation alone, or if the differences are due to other factors (skill, for instance).

**Problem D1**

The probability is one-half. If you look at the picture of the Push Penny board, you’ll notice that the shaded strips are one diameter wide, and the unshaded strips are also one diameter wide:

This means that, if the game is random, it is just as likely for the coin to land on a shaded strip as on an unshaded strip. This makes the probability of hitting a line (and landing on a shaded strip) equal to one-half, or 50%.

**Problem D2
**Let’s use a probability table to compare the experimental probability for this player to the probabilities for a random player:

This player seems to have improved. In particular, this player’s experimental probability of getting four hits in four tries is more than three times larger than the expected probability for a random player. This suggests that this player has developed skill in playing Push Penny.

**Problem D3**

Answers will vary. Good luck!