Explaining,the,Link,Between,Pa education Explaining the Link Between Pascals Triangle and Probabilit
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One famous pattern in Pascal's triangle is that if you toss n coins, thechance of getting m heads is the mth number along in the nth row of Pascal'striangle. In this article, I have attempted to explain this intriguing linkbetween Pascal's triangle and probability.As an example to help us understand why there is a link between Pascals'triangle and probability, let's take the situation of tossing 10 coins andtrying to get 6 heads. There are two ways that this can be done. We could tossthe first 9 coins and get 5 heads. Then we would have to get another head fromthe 10th coin to give us 6 heads out of 10 tosses. A second option is that wecould get 6 heads from our first 9 coin tosses, and then get a tail from our10th coin, again giving us 6 heads out of 10 tosses. Amazingly, simple as thisis, it is the essence of the proof of the link between Pascal's triangle andprobability!Think about what we have said so far: the number of ways of getting 6 headsout of 10 is the sum of the number of ways of getting 5 heads out of 9 and 6heads out of 9...All we need to do is think about where the situations discussed above appearin Pascal's triangle:1 9 36 84 126 126 84 36 9 11 10 45 120 210 252 210 120 45 10 1The numbers corresponding to 5/9 heads and 6/9 heads are 126 and 84, andfrom the rules of Pascal's triangle. we know that these will equal 210, as theyare (when the triangle is centralised) the two numbers directly above 210. Weknow that 6 out of 10 heads will be directly below 5 out of 9 and 6 out of 9heads, because the link we are trying to prove states the total number of cointosses shows you what row to look in, and the number of heads you're trying toget is one less than how many rows in you count. If you think about it, you getthe 9th row, 6th number in, and the 9th row, 7th number in, which will bepositioned directly above the 10th row, 7th number in if you centralise thetriangle.This argument is no different for getting any number of heads from anynumber of coin tosses. (Note: the only cases which are slightly different arewhen we are trying to get all heads or no heads. Then, thereis only one way of doing this, not two. For example, when trying to get 0 headsfrom 5 coin tosses, the only way of achieving this is to get 0 heads from 4coin tosses, so there is no addition of two different values. This is just likein Pascal's triangle, however, when you're at the edge calculating the 1s, andonly adding a single number (a one) from above).Essentially, what we have shown is that assuming this link between Pascal'striangle and probability works for the nth row, we can show it also works forthe (n+1)th row. The rule clearly works for row 1 (you can check this), andtherefore works for row 2, and as it works for row 2, we have shown it mustwork for row 3. We can carry on this reasoning indefinitely, showing the ruleworks for any row of Pascal's triangle (this method of proof is calledproof by induction).Therefore, we have shown our desired result, and it is (hopefully!) clearwhy Pascal's triangle is so inextricably linked the tossing of coins and therepetition of similar equal probabilities. 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;}
Explaining,the,Link,Between,Pa