### Theoretical and Experimental Coin Tosses

In an example in a previous concept, 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. Press \begin{align*}\boxed{\text{APPS}}\end{align*} and choose Prob Sim to see these simulations.

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you will have the following screen appear, with Toss Coins as the first option.

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.

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

#### Using the Toss Coins Simulation

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

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*}

#### Determining Experimental Probability

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

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 A, 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.

2. 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?

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*}

**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?

### Examples

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

#### Example 1

Determine the theoretical probability for the teacher.

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 factorial function from an earlier concept.

*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} & = 300 3\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.

#### Example 2

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

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*}

### Review

- Use the randBin function on your calculator to simulate tossing 5 coins 25 times to determine the probability of getting 2 tails.
- Use the randBin function on your calculator to simulate tossing 10 coins 50 times to determine the probability of getting 4 heads.
- Calculate the theoretical probability of getting 3 heads in 10 tosses of a coin.
- Find the experimental probability using technology of getting 3 heads in 10 tosses of 3 coins.
- Calculate the theoretical probability of getting 8 heads in 12 tosses of a coin.
- Calculate the theoretical probability of getting 7 heads in 14 tosses of a coin.

3 coins were tossed 500 times using technology.

- According to the following screen, what is the experimental probability of getting 0 heads?
- According to the following screen, what is the experimental probability of getting 1 head?
- According to the following screen, what is the experimental probability of getting 2 heads?
- According to the following screen, what is the experimental probability of getting 3 heads?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.6.