## InterA.CoinToss2of5 History

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\*What "identifies" the combinations with probability: (0.4)^4 \cdot (0.6)^1 ?

- What "identifies" the combinations with probability: (0.4)^4 \cdot (0.6)^1 ?

\*What "identifies" the combinations with probability: (0.4)^4 /cdot (0.6)^1 ?

\*What "identifies" the combinations with probability: (0.4)^4 \cdot (0.6)^1 ?

- What "identifies" the combinations with probability: (0.4)^4 /cdot (0.6)^1 ?

\*What "identifies" the combinations with probability: (0.4)^4 /cdot (0.6)^1 ?

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

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

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.

Answers to Guided Activities

Answers to Guided Activities

Answers to Guided Activities

Answers to Guided Activities

Answers to Guided Activities

Do the experiment once (Click ). Do you understand what a winning combination is?

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

Do the experiment 1000 times (Click ).

Do the experiment 1000 times by clicking .

Change the weight of getting a heads, run the experiment 1000 times, calculate the theoretical probability and compare your results.

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

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

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

Do the experiment 1000 times (Click ).

- What is the percent of Wins?
- What is the percent of Losses?

Do the experiment 1000 times (Click ).

- What is the percent of Wins?
- What is the percent of Losses?

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

Do the experiment once (Click ). Do you understand what a winning combination is?

Do the experiment 1000 times.

Do the experiment 1000 times (Click ).

- How many different combinations are there of this type (i.e. with this probability)? Think combinatorics ...

- How many different combinations are there of this type (i.e. with this probability)? Think combinatorics.

(a) Win = 2 or more Heads in 5 tosses.

(a) Win = 2 or more Heads in 5 tosses.

(b) Lose = less than 2 Heads in 5 tosses.

(b) Lose = less than 2 Heads in 5 tosses.

[ valign=middle]***Click on the Toss button for a single trial or the 1000 button for 1000 trials

[ valign=middle]***Click on for a single trial or for 1000 trials.

Example: What is the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72 ?

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

- Change the number of trials by changing the value of Number Coins from 5 to 10.
- Change the number of success required by changing the value of Number of Heads to Win from 2 to 8.

- 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.

- 2 Balls - 1 Red and 1 Green Drop into Vertical Box 3rd grade probability

- Mathcast of Problem: Theoretical probability of 3 out of 10

[ valign=middle]***[]Click on the Toss button for a single trial or the 1000 button for 1000 trials

[ valign=middle]***Click on the Toss button for a single trial or the 1000 button for 1000 trials

[ valign=middle]*** Weight the coin for Heads using the slider for p.

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[]Click on the Toss button for a single trial or the 1000 button for 1000 trials

[ valign=middle]***[]Click on the Toss button for a single trial or the 1000 button for 1000 trials

To use offline, download and unzip 6balls_3R_3G_3Boxes.zip.

Then, double-click on balls_3R_3G_3Boxes.html.To use offline, download and unzip CoinToss2of5.zip.

Then, double-click on CoinToss2of5.html.Suppose you want to know the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

Example: What is the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72 ?

(a) Win = 2 or more Heads in 5 tosses.

We label this event: 0

(a) Win = 2 or more Heads in 5 tosses.

(b) Lose = less than 2 Heads in 5 tosses.

We label this event: 1

(b) Lose = less than 2 Heads in 5 tosses.

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

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

- Which should be easier to calculate: Theoretical probability of winning or losing?

- Here, which should be easier to calculate - the theoretical probability of winning or losing?

Bernoulli experiments*

Suppose you want to know the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

Suppose you want to know the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

A binomial trial or *Bernoulli trial* has exactly 2 mutually exclusive events W and L with p=Pr(W).

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

Other Bernoulli trials

Suppose you want to know the probability of k=3 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

Bernoulli experiments*

Suppose you want to know the probability of k=8 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

- Change the number of success required by changing the value of Number of Heads to Win from 2 to 3.

- Change the number of success required by changing the value of Number of Heads to Win from 2 to 8.

- *Does your theory match your experiment?

- 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?

This is a binomial trial* where

In this experiment we have decided that:

Suppose you want to know the probability of k=3 or more successes in n=10 trials of a Bernoulli experiment with p=0.72

- Change the number of trials by changing the value of Number Coins from 5 to desired number.
- Change the number of success required by changing the value of Number of Heads to Win from 2 to desired number.
- Click and drag the slider p to the desired weight, i.e the probability of getting a Heads.

- Change the number of trials by changing the value of Number Coins from 5 to 10.
- Change the number of success required by changing the value of Number of Heads to Win from 2 to 3.
- Click and drag the slider p to 72.
- Run the experiment at least 1000 times.

*Does your theory match your experiment?

- *Does your theory match your experiment?