Session 8, Part D:
Are You a Random Player?

In This Part: Developing the Mathematical Probability | Comparing Mathematical and
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. 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:

Number of Hits

Frequency

Probability
(Fraction)

Probability
(Decimal)

Probability
(Percentage %)

 0 1 1/16 .0625 6.25% 1 4 4/16 .25 25% 2 6 6/16 .375 37.5% 3 4 4/16 .25 25% 4 1 1/16 .0625 6.25%

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

Number of Hits

Experimental Frequency

Experimental Probability

Probability for Random Player

 0 5 .05 .0625 1 22 0.22 .25 2 36 0.36 .375 3 29 0.29 .25 4 8 0.08 .0625

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?

Number of Hits

Experimental
Frequency

 0 2 1 14 2 29 3 34 4 21

 Problem D3 Use your data from Problem A5 to determine whether you were developing any skill for Push Penny. Then play another 20 times to see if your skill has improved over the course of this session.

 Video Segment In this video segment, Doug McCrum explains how his company, Global Specialty Risk, uses probability models to analyze the risks involved in clients' promotional contests. As you watch, take note of how probability and statistics are used by Doug McCrum's company. If you're using a VCR, you can find this segment on the session video approximately 21 minutes and 35 seconds after the Annenberg Media logo.

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 Session 8: Index | Notes | Solutions | Video