Quiz

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Section

Section 1 (9-10am)Section 2 (10-11am)

Estimating Probability

Simulating a random process

Let's turn to the computer to repeat this random process many many times.

• Press Randomize. Notice that the applet randomly returns babies to mothers and determines how many babies are returned to the correct home (by matching diaper colors). The applet also counts and graphs the resulting number of matches.

• Uncheck the Animate box and press Randomize a few more times. You should see the results accumulate in the table and the histogram.

• Click on the histogram bar representing the outcome of zero mothers receiving the correct baby. The graph on the right now shows a “time plot” of the proportion of trials with 0 matches vs. the number of trials.

• Set the Number of trials to 100 and press Randomize a few times, noticing how the behavior of this graph changes.

(i) Describe what you learn about how the (cumulative) proportion of trials that result in zero matches changes as you simulate more and more trials of this process.

(j) After at least 1000 trials, complete the table below for the Number of Matches.

	0	1	2	3	4	Total
Count
Proportion

(k) Based on your answers to (j), what is your empirical estimate for the probability that at least one mother receives her own baby?
Did all of your classmates obtain the same estimate?

YesNo
(l) Return to the applet above and check the Show Theoretical button. Describe how your empirical (simulated-based) estimate of the probability of 0 matches compares to the theoretical estimate. (Hint: Your answer may change depending on where in the time plot you are looking?)
(m) Save a screen capture of your Relative Frequency graph (snipping tool or right-click and Save Image As?) and upload your image here.

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Interpreting Probability

In this course we will use the "long-run proportion" definition of probability to determine and make sense of probabilities.

Definition: The probability of an outcome refers to the long-run proportion of times that the outcome would occur if the random process were repeated a very large number of times under identical conditions. You can approximate a probability by simulating the process many times. Simulation leads to an empirical estimate of the probability, which is the proportion of times that the event occurs in the simulated repetitions of the random process.

When interpreting a probability statement, you should be able to identify

i. what random process is being repeated over and over again (what are the identical conditions) and

ii. what proportion is being calculated (e.g. proportion of wins).

Your interpretation should not include the words “probability,” “chance,” "odds," or “likelihood" or any other synonyms of "probability."

For example, if the probability of getting a red M&M candy is 0.20, we would interpret that as saying if we were to repeatedly select an M&M candy over and over forever from an infinitely large bag (!), then, in the long run, about 20% of the selected M&Ms will be red.

Using this approach, how should we interpret the following statements:

1. The probability a randomly selected American wants to retain the penny as the lowest unit of U.S. currency is 0.70.
2. The probability of winning at a ‘daily number’ lottery game is 1/1000.[Hint: Your answer should not include the number 1000!]
3. There is a 30% chance of rain tomorrow. [Hint: Follow the rules above!]
Should be Empty:

Estimating Probability

Interpreting Probability