# 3.4: Using Technology to Find Probability Distributions

**At Grade**Created by: CK-12

If we look back at Example 1, we were tossing 2 coins. If you were to repeat this experiment 100 times, or if you were going to toss 10 coins 50 times, these experiments would be very tiring and take a great deal of time. On the TI-84 calculator, there are applications built in to determine the probability of such experiments. In this section, we will look at how you can use your graphing calculator to calculate probabilities for larger trials and draw the corresponding histograms.

On the TI-84 calculator, there are a number of possible simulations you can do. You can do a coin toss, spin a spinner, roll dice, pick marbles from a bag, or even draw cards from a deck.

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}

Let’s try a spinner problem. Choose Spin Spinner.

*Example 9*

You are spinning a spinner like the one shown below 20 times. How many times does it land on blue?

*Solution:*

In Spin Spinner, a wheel with 4 possible outcomes is shown. You can adjust the number of spins, graph the frequency of each number, and use a table to see the number of spins for each number. Let’s try this problem. We want to set this spinner to spin 20 times. Look at the keystrokes below and see how this is done.

In order to match our color spinner with the one found in the calculator, you will see that we have added numbers to our spinner. This is not necessary, but it may help in the beginning to remember that 1 = blue (for this example).

Now that the spinner is set up for 20 trials, choose SPIN by pressing \begin{align*}\boxed{\text{WINDOW}}\end{align*}

We can see the result of each trial by choosing TABL, or pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*}

And we see the graph of the resulting table, or go back to the first screen, simply by choosing GRPH, or pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*}

Now, the question asks how many times we landed on blue (number 1). We can actually see how many times we landed on blue for these 20 spins. If you press the right arrow \begin{align*}\left ( \boxed{\blacktriangleright} \right )\end{align*}, the frequency label will show you how many of the times the spinner landed on blue (number 1).

To go back to the question, how many times does the spinner land on blue if it is spun 20 times? The answer is 3. To calculate the probability of landing on blue, we have to divide by the total number of spins.

\begin{align*}P(\text{blue}) = \frac{3}{20} = 0.15\end{align*}

Therefore, for this experiment, the probability of landing on blue with 20 spins is 15%.

The above example introduces us to a new concept. We know that the spinner has 4 equal parts (blue, purple, green, and red). In a single trial, we can assume that:

\begin{align*}P(\text{blue}) = \frac{1}{4} = 0.25\end{align*}

However, we know that we did the experiment and found that the probability of landing on blue, if the spinner is spun 20 times, is 0.15. Why the difference?

The difference between these 2 numbers has to do with the difference between theoretical and experimental probability. **Theoretical probability** is defined as the number of desired outcomes divided by the total number of outcomes.

**Theoretical Probability**

\begin{align*}P(\text{desired}) = \frac{\text{number of desired outcomes}}{\text{total number of outcomes}}\end{align*}

For our spinner example, the theoretical probability of landing on blue is 0.25. Finding the theoretical probability requires no collection of data.

In the case of the experiment of spinning the spinner 20 times, the probability of 0.15, found by counting the number of times the spinner landed on blue, is called the experimental probability. **Experimental probability** is, just as the name suggests, dependent on some form of data collection. To calculate the experimental probability, divide the number of times the desired outcome has occurred by the total number of trials.

**Experimental Probability**

\begin{align*}P(\text{desired}) = \frac{\text{number of times desired outcome occurs}}{\text{total number of trials}}\end{align*}

You can try a lot of examples and trials yourself using the NCTM Illuminations page found at http://illuminations.nctm.org/activitydetail.aspx?ID=79.

What is interesting about theoretical and experimental probabilities is that, in general, the more trials you do, the closer the experimental probability gets to the theoretical probability. To show this, try spinning the spinner for the next example.

*Example 10*

You are spinning a spinner like the one shown below 50 times. How many times does it land on blue?

*Solution:*

Set the spinner to spin 50 times and choose SPIN by pressing \begin{align*}\boxed{\text{WINDOW}}\end{align*}.

You can see the result of each trial by choosing TABL, or pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*}.

Again, we can see the graph of the resulting table, or go back to the first screen, simply by choosing GRPH, or pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*} again.

The question asks how many times we landed on blue (number 1) for the 50 spins. Press the right arrow \begin{align*}\left ( \boxed{\blacktriangleright} \right )\end{align*}, and the frequency label will show you how many of the times the spinner landed on blue (number 1).

Now go back to the question. How many times does the spinner land on blue if it is spun 50 times? The answer is 11. To calculate the probability of landing on blue, we have to divide by the total number of spins.

\begin{align*}P(\text{blue}) = \frac{11}{50} = 0.22\end{align*}

Therefore, for this experiment, the probability of landing on blue with 50 spins is 22%.

If we tried 100 trials, we would see something like the following:

In this case, we see that the frequency of 1 is 23.

So how many times does the spinner land on blue if it is spun 100 times? The answer is 23. To calculate the probability of landing on blue in this case, we again have to divide by the total number of spins.

\begin{align*}P(\text{blue}) = \frac{23}{100} = 0.23\end{align*}

Therefore, for this experiment, the probability of landing on blue with 100 spins is 23%. You can see that as we perform more trials, we get closer to 25%, which is the theoretical probability.

*Example 11*

How many times do you predict we would have to spin the spinner in Example 10 to have the experimental probability equal the theoretical probability?

*Solution:*

With 170 spins, we get a frequency of 42 for blue.

The experimental probability in this case can be calculated as follows:

\begin{align*}P(\text{blue}) = \frac{42}{170} = 0.247\end{align*}

Therefore, the experimental probability is 24.7%, which is even closer to the theoretical probability of 25%. While we're getting closer to the theoretical probability, there is no number of trials that will guarantee that the experimental probability will exactly equal the theoretical probability.

Let’s try an example using the Toss Coins simulation.

*Example 12*

A fair coin is tossed 50 times. What is the theoretical probability and the experimental probability of tossing tails on the fair coin?

*Solution:*

To calculate the theoretical probability, we need to remember that the probability of getting tails is \begin{align*}\frac{1}{2}\end{align*}, or:

\begin{align*}P(\text{tails}) = \frac{1}{2} = 0.50\end{align*}

To find the experimental probability, we need to run the Toss Coins simulation in the probability simulator. We could also actually take a coin and flip it 50 times, each time recording if we get heads or tails.

If we follow the same keystrokes to get into the Prob Sim app, we get to the main screen.

Choose Toss Coins and then choose SET by pressing \begin{align*}\boxed{\text{ZOOM}}\end{align*}.

Choose OK by pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*} and go back to the main screen. Then choose TOSS by pressing \begin{align*}\boxed{\text{WINDOW}}\end{align*}.

To find the frequency, we need to press the \begin{align*}\boxed{\blacktriangleright}\end{align*} arrow to view the frequency for the tossing experiment.

We see the frequency of tails is 30. Now we can calculate the experimental probability.

\begin{align*}P(\text{tails}) = \frac{30}{50} = 0.60\end{align*}

*Example 13*

What if the fair coin is tossed 100 times? What is the experimental probability? Is the experimental probability getting closer to the theoretical probability?

*Solution:*

To find the experimental probability for this example, we need to run the Toss Coins simulation in the probability simulator again. You could also, like in Example 12, actually take a coin and flip it 100 times, each time recording if you get heads or tails. You can see how the technology is going to make this experiment take a lot less time.

Choose Toss Coins and then choose SET by pressing \begin{align*}\boxed{\text{ZOOM}}\end{align*}.

Choose OK by pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*} and go back to the main screen. Then choose TOSS by pressing \begin{align*}\boxed{\text{WINDOW}}\end{align*}.

To find the frequency, we need to press the \begin{align*}\boxed{\blacktriangleright}\end{align*} arrow to view the frequency for the tossing experiment.

Notice that the frequency of tails is 59. Now you can calculate the experimental probability.

\begin{align*}P(\text{tails}) = \frac{59}{100} = 0.59\end{align*}

With 50 tosses, the experimental probability of tails was 60%, and with 100 tosses, the experimental probability of tails was 59%. This means that the experimental probability is getting closer to the theoretical probability of 50%.

You can also use this same program to toss 2 coins or 5 coins. Actually, you can use this simulation to toss any number of coins any number of times.

*Example 14*

2 fair coins are tossed 10 times. What is the theoretical probability of both coins landing on heads? What is the experimental probability of both coins landing on heads?

*Solution:*

The theoretical probability of getting heads on the first coin is \begin{align*}\frac{1}{2}\end{align*}. Flipping the second coin, the theoretical probability of getting heads is again \begin{align*}\frac{1}{2}\end{align*}. The overall theoretical probability is \begin{align*}\left ( \frac{1}{2} \right )^2\end{align*} for 2 coins, or:

\begin{align*}P(2 H) & = \frac{1}{2} \times \frac{1}{2}\\ P(2H) & = \left ( \frac{1}{2} \right )^2\\ P(2H) & = \frac{1}{4}\end{align*}

To determine the experimental probability, let’s go to the probability simulator. Again, you can also do this experiment manually by taking 2 coins, tossing them 10 times, and recording your observations.

Choose Toss Coins and then choose SET by pressing \begin{align*}\boxed{\text{ZOOM}}\end{align*}.

Choose OK by pressing \begin{align*}\boxed{\text{GRAPH}}\end{align*} and go back to the main screen. Then choose TOSS by pressing \begin{align*}\boxed{\text{WINDOW}}\end{align*}.

Find the frequency of getting 2 heads \begin{align*}(2 H)\end{align*}.

The frequency is equal to 4. Therefore, for 2 coins tossed 10 times, there were 4 times that both coins landed on heads. You can now calculate the experimental probability.

\begin{align*}P(2 H) & = \frac{4}{10}\\ P(2 H) & = 0.40 \ \text{or} \ 40\%\end{align*}

To try other types of probability simulations, you can use the Texas Instruments Activities Exchange. Look up simple probability simulations on http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=9327.

You can also use the randBin function on your calculator to simulate the tossing of a coin. The randBin function is used to produce experimental values for discrete random variables. You can find the randBin function using:

\begin{align*}\boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \boxed{\blacktriangledown} \ \boxed{\blacktriangledown} \ \boxed{\blacktriangledown} \ \boxed{\blacktriangledown} \ \boxed{\blacktriangledown} \ \boxed{\blacktriangledown} \ (\boxed{7})\end{align*}

If you wanted to toss 4 coins 10 times, you would enter the command below:

The list that is produced contains the count of heads resulting from each set of 4 coin tosses. If you use the right arrow \begin{align*}\left ( \boxed{\blacktriangleright} \right )\end{align*}, you can see how many times from the 10 trials you actually had 4 heads.

*Example 15*

You are in math class. Your teacher asks what the probability is of obtaining 5 heads if you were to toss 15 coins.

(a) Determine the theoretical probability for the teacher.

(b) Use the TI calculator to determine the actual probability for a trial experiment of 10 trials.

*Solution:*

(a) Let’s calculate the theoretical probability of getting 5 heads in the 15 tosses. In order to do this type of calculation, let’s bring back the concept of factorial from an earlier lesson.

*Numerator (Top)*

In the example, you want to have 5 H's and 10 T‘s. Our favorable outcomes would be HHHHHTTTTTTTTTT, with the H's and T's coming in any order. The number of favorable outcomes would be:

\begin{align*}\text{number of favorable outcomes} & = \frac{\text{number of tosses!}}{\text{number of heads!} \times \text{number of tails!}}\\ \text{number of favorable outcomes} & = \frac{15!}{5! \times 10!}\\ \text{number of favorable outcomes} & = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(5 \times 4 \times 3 \times 2 \times 1) \times (10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}\\ \text{number of favorable outcomes} & = \frac{1.31 \times 10^{12}}{120 \times 3628800}\\ \text{number of favorable outcomes} & = 3003\end{align*}

*Denominator (Bottom)*

The number of possible outcomes \begin{align*}= 2^{15}\end{align*}

The number of possible outcomes = 32,768

Now you just divide the numerator by the denominator:

\begin{align*}P (5 \text{ heads}) & = \frac{3003}{32768}\\ P(5 \text{ heads}) & = 0.0916\end{align*}

Therefore, the theoretical probability would be 9.16% of getting 5 heads when tossing 15 coins.

b) To calculate the experimental probability, let’s use the randBin function on the TI-84 calculator.

From the list, you can see that you only have 5 heads 1 time in the 10 trials.

Therefore, the experimental probability can be calculated as follows:

\begin{align*}P(5 \ \text{heads}) = \frac{1}{10} = 10\%\end{align*}

**Points to Consider**

- How is the calculator a useful tool for calculating probabilities in discrete random variable experiments?
- How are these experimental probabilities different from what you would expect the theoretical probabilities to be? When can the 2 types of probability possibly be equal?

**Vocabulary**

- Binomial distribution
- A distribution produced by an experiment with 2 possible outcomes, where there is a fixed number of successes in \begin{align*}X\end{align*} (random variable) trials, and each trial is independent of the others.

- Discrete random variables
- Only have a specific (or finite) number of numerical values within a certain range.

- Experimental probability
- The actual probability of an event resulting from an experiment.

- Factorial function (!)
- The function of multiplying a series of consecutive descending natural numbers.

- Histogram
- A graph that uses vertically arranged bars to display data.

- Multinomial distribution
- A distribution produced by an experiment where the number of possible outcomes is greater than 2 and where each outcome has a specific probability.

- Probability distribution
- A table, a graph, or a chart that shows you all the possible values of \begin{align*}X\end{align*} (your variable), and the probability associated with each of these values \begin{align*}(P(X))\end{align*}.

- Random variables
- Variables that take on numerical values governed by a chance experiment.

- Theoretical probability
- A probability that is the ratio of the number of different ways an event can occur to the total number of equally likely possible outcomes. The numerical measure of the likelihood that an event, \begin{align*}E\end{align*}, will happen.

\begin{align*}P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\end{align*}

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