Pascal,Triangle,and,Probabilit education Pascals Triangle and Probability
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Of all the patterns and discoveries Blaise Pascal made from examiningPascal's triangle, it was perhaps its link with probability that made thetriangle so interesting to him and other mathematicians of his time. In thisarticle, I discuss how Pascal's triangle can be used to calculate probabilitiesconcerned with the tossing of coins (or similar 50:50 actions) repeated anumber of times.To discover this hidden link between Pascal's triangle and probability, wecan begin by looking at the different combinations that can be made fromtossing 1,2 and 3 coins.When just one coin is tossed, there are clearly just two outcomes, each withan equal chance of occurring. These can be represented as H and T. However,when two coins are tossed, there are four outcomes: TT, TH, HT and HH. It isimportant to distinguish between HT and TH - we must class a head from Coin 1and a tail from coin 2 as a different combination to a tail from coin 2 and ahead from coin 1. (Incidentally, this is where a lot of the mathematicians inPascal's time went wrong - they treated HT the same as TH, and so ended up withincorrect probabilities).With three coins, there are 8 different outcomes. They are HHH, HHT, HTH,THH, TTH, THT, HTT and TTT. This is summarised below:3 Head: HHH - 1 way2 Heads: HHT, HTH or THH - 3 ways1 Head: HTT, THT or TTH - 3 ways0 Heads: TTT - 1 wayAs each time a fair coin is tossed, there is an equal chance of both headsand tails, each combination listed above is equally likely. This gives us atotal of 8 equal outcomes. Therefore, to calculate the probability, all we needto do it divide the number of combinations by 8, giving the probabilities 1/8 =12.5% for 0 and 3 heads, and 3/8 = 37.5% for 1 and 2 heads.If you look at the information above, you can also see that there is only 1way of getting 0 or 3 heads, but 3 ways of getting 1 or 2 heads. It should notbe too surprising that there are more ways of getting 1 or 2 heads, resultingin a higher probability of these totals, as you would expect to get headsroughly half of the time. Obviously with 3 coin tosses, you can't get half ofthem heads, but it makes sense that the closer you get to this halfway mark,the higher the probability of that outcome occurring.That's enough chatter now. Let's get on with the interesting stuff. How doesall this link in with Pascal's triangle?! Well, the numbers in the table above(in the "number of ways" column) are 1,3,3,1. This is the third rowof Pascal's triangle! If you create similar tables for one and two coin tosses,you should get 1,1 and 1,2,1, which are the first and second rows of Pascal'striangle.This is very exciting! What it means is that we can use Pascal's triangle tocalculate probabilities in seconds that would have otherwise taken hours. Forexample, consider this question: If I toss 10 coins, what is the chance thatexactly 6 of them will be heads?We need to look at the 6th number in on the 10th row of Pascal's triangle.It is 210. And a quick calculation tells us that the total of all the numbersin row 10 is 1024. Before you can blink, we have calculated that theprobability is 210/1024, or about 21%. Now, you've got to admit that was muchquicker than writing out all 1024 combinations wasn't it? Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;}
Pascal,Triangle,and,Probabilit