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(:title Coin Toss Activity: 2 out of 5 - Advanced Placement Probability:) [table border=1 cellpadding=5 width=100%] [row] [c] [table border=0] [row] [ valign=middle]%c11 exa%***Click on Attach:toss.png for a single trial or Attach:1000.png for 1000 trials. [tableend] ---- (:html:) (:htmlend:) >>&<< %sol%%popwin id=1 left=0 top=50 width=700 height=520%[[http://mathcasts.org/mtwiki/uploads/InterA/CoinToss2of5.pdf|Printable: Guided Activities]] >>&<< %reg%%popwin id=2 left=300 top=100 width=700 height=520%[[http://mathcasts.org/mtwiki/uploads/InterA/CoinToss2of5_Answers.pdf|Printable: Answers to Guided Activities]]%%%% To use offline, download and unzip [[http://mathcasts.org/mtwiki/uploads/InterA/CoinToss2of5.zip|CoinToss2of5.zip]]. >>&<>&<<[[http://mathcasts.org/mc/scratch/how2offline/how2offline.htm|%ref%How do I do all this?%%]] Scratch file designed by [[mailto:emath@emathforall.com?subject=scratch|Linda Fahlberg-Stojanovska]]. [ valign=middle] (:div222 class=s9 style="margin-left:3px":) %c12 mor b%Activity Notes%% >>&<< ---- Our coin is "weighted" so that p(Heads)=0.4 and we toss it 5 times. %c11 exa%What is the probability that at least 2 of the tosses are heads?? >>&<< ---- >>&<< In this experiment we have decided that: (a) %pra%Win%% = %pra%2 or more Heads in 5 tosses%%. (b) %red%Lose%% = %red%less than 2 Heads in 5 tosses%%. >>&<< ---- (:showhide init=show div=div1 lshow="+" lhide="-":) %c11 exa%Guided Activity 1:%% 2 out of 5%% (:div1 id=div1 :) >>&<< %c11 reg%Do the experiment once by clicking Attach:toss.png . Do you understand what a winning combination is?%% >>&<< %c11 sol%Do the experiment 1000 times by clicking Attach:1000.png .%% * What is the percent of %pra%Wins%%? * What is the percent of %red%Losses%%? * Here, which should be easier to calculate - the theoretical probability of winning or losing? >>-<< %sol%Now for some dreaded theory ...%% >>&<< %bgr%1. An example of a winning combination is TTHHT. What is the probability of getting this combination? >>&<< %def%2. Another example of a winning combination is HHTHT. What is the probability of getting this combination? >>&<< %mor%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 $}? *How many different combinations are there of this type (i.e. with this probability)? Think combinatorics. >>-<< %exa%Losing combinations *How many '''types''' of losing combinations are there? *What is the '''probability of each of these types''' of losing combinations? *%exa%What is the probability of losing?%% >>&<< %pra%Winning combinations%% *%pra%What is the probability of winning?%% *Does this correspond to your empirical results when you ran the experiment 1000 times? >>&<< ---- >>&<< %c11 ans%Change the %pink%weight%% of getting a heads, click Attach:1000.png , calculate the theoretical probability and compare your results.%% >>&<< ---- (:div1end:) (:showhide init=show div=div2 lshow="+" lhide="-":) %c11 exa%Guided Activity 2:%% Other Bernoulli* experiments%% (:div2 id=div2 :) >>&<< %c11 bgr%Example:%% What is the probability of %red%{$ k=12 $}%% or more successes in %ans%{$ n=15 $}%% trials of a Bernoulli experiment with %pink%{$ p=0.72 $}%%? *Download the zip and open the scratch file CoinToss2of5(you must have scratch installed). *Click on the '''3rd''' sprite TossM (1000). *In the script %mor%''When TossM clicked''%%: ** Change the number of trials by changing the value of %ans%Number Coins%% from 5 to 15. ** Change the number of success required by changing the value of %red%Number of Heads to Win%% from 2 to 12. ** Click and drag the slider p to 72. ** Run the experiment at least 1000 times. >>&<< *Calculate the theoretical probability of winning. >>&<< --> Recall: The theoretical probability of getting exactly %red%k%% successes in %ans%n%% trials with a probability %pink%p%% of success in any individual trial is{$ 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 %red%k%%? >>&<< *%c12 exa%*Does your theory match your experiment?%% >>&<< (:div2end :) ---- (:div class="s9" style="margin-left:10px":) %sol%A %red%[[http://en.wikipedia.org/wiki/Bernoulli_trial|''Bernoulli or Binomial Trial'']]%% has exactly 2 events: %pra%Success (Win)%% and %red%Fail (Lose)%% with %pink%p=Pr(Success)%%.%% (:divend:) (:div222end:) [tableend] >>-<< * To change Scratch file, download and install freeware Scratch. See [[http://scratch.mit.edu|Scratch-MIT]] for more information. >>&<< %b ref%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. ---- >>&<< %rel b%Related Topics%% * %popwin id=1 left=0 top=50 width=700 height=520%[[http://mathcasts.org/gg/enliven/ps/ap/CoinToss3of10/CoinToss3of10.html|Mathcast of Problem: Theoretical probability of 3 out of 10]] * [[DiceDifference|Throw two dice and check their difference]] %sol s9%6th grade probability%% * [[4Balls2Boxes|4 Balls - 2 Reds and 2 Greens Drop into 2 Boxes]] %sol s9%8th grade probability%% >>&<< ---- >><< [table width=100% border=0] [row] [][[activity/| Attach:main/tri_purple_up_a.gif ]] [[activity/| Up one level]] [r](:html:) (:htmlend:) [tableend]