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

 ***Click on for a single trial or for 1000 trials.

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?