Bernoulli Trials and Binomial Experiments Math 171 Lab #1

Introduction

In class you tossed 4 coins to see how many heads would appear. When you graphed everyone's results together, you saw the distribution of the number of heads for several such tosses. This graph, based on experimental results, approximates the theoretical distribution of the number of heads in 4 tosses of a fair coin. To see an even better estimate of this distribution, you should toss the 4 coins 100 (or, better, 1000) times... but that would be time-consuming, and probably pretty boring. Let's simulate the process with DataDesk.

First, you need to know the official terminology and notation.

Tossing a coin is one example of a Bernoulli Trial.

• There are only two outcomes, heads and tails. These are commonly called "Success" and "Failure".
• The probability of success (denoted p) is constant from trial to trial.

In a Binomial Experiment we repeat independent, identical Bernoulli Trials some number of times (denoted n), and define a random variable X = the number of Successes (e.g heads). You will be studying binomial probabilities next week in class.

Now we will simulate this binomial experiment 100 times (cases).

• Under Manip choose Generate random numbers.
• Specify 1 random variable with 100 cases.
• Select Binomial Experiment
• Specify 4 Bernoulli trials per experiment, with probability of success = 0.5.
• OK and DataDesk does it, displaying an icon for the random variable.
• Double-click on the variable icon to see the values of the random variable. (There should be 100 results between 0 and 4, indicating how many heads appeared each time the 4 coins were tossed).
• Plot Histogram ‡ to graph the number of times in the 100 simulations we ended up with 0, 1, 2, 3, or 4 heads.
• You may need to fix the graph so the bars make sense. Under Modify choose Scale, Plot scale and specify to align bars at whole number (here 0) with bar width = 1. You may want to have 1 bar per tickmark (label) on the x - axis this time, but often 5 is best.

Got it? OK, you're ready to do the assignment.

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Exercises

1. Graph ‡ the distribution of the number of heads you might see if you tossed 100 coins.
   1. Run 50 simulations (cases).
   2. Write a sentence or two describing the result.
2. Now consider a different binomial situation: a multiple choice test, each question having 4 choices. You do not know any of the answers and simply guess at every one.
   1. Suppose the test has 7 questions. Describe this situation. (What is one Bernoulli trial? What is the probability of success? How many trials in this binomial experiment? Define an appropriate random variable.)
   2. How many might you get right by guessing? Simulate this experiment 100 times, and graph ‡ the distribution.
   3. Describe what the graph shows you. Compare this graph to the first one we did involving the tossing of 4 coins. How are they different? Do you have a reasonable chance to get a respectable grade (3 or 4 right) by guessing?
   4. Now suppose the test has 50 of these 4 - choice questions. Simulate taking such a test with 100 cases, and describe what the result shows this time. ‡ (Notice that in theory the x - axis runs from 0 to 50 possible correct answers, but this full range probably did not happen in your simulation. DataDesk focuses on what actually occurred.

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To turn in:

‡ Please:

• Print out the results marked with ‡ and write answers to the questions posed in the Exercises.
• Hand in your completed assignment at the start of lab next week. ‡