Coin Toss Activity: 2 out of 5 - Advanced Placement Probability

***Click on

for a single trial or

for 1000 trials.

Printable: Guided Activities

Printable: Answers to Guided Activities

To use offline, download and unzip CoinToss2of5.zip.

Then, double-click on CoinToss2of5.html.

How do I do all this?

Scratch file designed by Linda Fahlberg-Stojanovska.

Activity Notes

Our coin is "weighted" so that p(Heads)=0.4 and we toss it 5 times.
What is the probability that at least 2 of the tosses are heads??

In this experiment we have decided that:
(a) Win = 2 or more Heads in 5 tosses.
(b) Lose = less than 2 Heads in 5 tosses.

Guided Activity 1: 2 out of 5

Do the experiment once by clicking . Do you understand what a winning combination is?

Do the experiment 1000 times by clicking .

What is the percent of Wins?
What is the percent of Losses?
Here, which should be easier to calculate - the theoretical probability of winning or losing?

Now for some dreaded theory ...

1. An example of a winning combination is TTHHT. What is the probability of getting this combination?

2. Another example of a winning combination is HHTHT. What is the probability of getting this combination?

3. A third example is HTTHH. This combination is "similar to" example 1 or example 2? Why?

What "identifies" the combinations with probability: (0.4)^4 \cdot (0.6)^1 ?
How many different combinations are there of this type (i.e. with this probability)? Think combinatorics.

Losing combinations

How many types of losing combinations are there?
What is the probability of each of these types of losing combinations?
What is the probability of losing?

Winning combinations

What is the probability of winning?
Does this correspond to your empirical results when you ran the experiment 1000 times?

Change the weight of getting a heads, click , calculate the theoretical probability and compare your results.

Guided Activity 2: Other Bernoulli* experiments

Example: What is the probability of k=12 or more successes in n=15 trials of a Bernoulli experiment with p=0.72 ?

Download the zip and open the scratch file CoinToss2of5(you must have scratch installed).
Click on the 3rd sprite TossM (1000).
In the script When TossM clicked:
- Change the number of trials by changing the value of Number Coins from 5 to 15.
- Change the number of success required by changing the value of Number of Heads to Win from 2 to 12.
- Click and drag the slider p to 72.
- Run the experiment at least 1000 times.

Calculate the theoretical probability of winning.

Recall: The theoretical probability of getting exactly k successes in n trials with a probability p of success in any individual trial is Pr (k,n,p) = \left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array}} \right){p^k}{\left( {1 - p} \right)^{n - k}} = \frac{{n!}}{{k!\left( {n - k} \right)!}}{p^k}{\left( {1 - p} \right)^{n - k}}

Think before calculating: What are the possible values for k?

*Does your theory match your experiment?

A Bernoulli or Binomial Trial has exactly 2 events: Success (Win) and Fail (Lose) with p=Pr(Success).

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MetaData: CA_APSP_3.0 Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.