document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207647>Question 1207647</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A high school baseball player has a 0.216 batting average. In one game, he gets 8 at bats. What is the probability he will get at least 4 hits in the game?<BR>\n" );
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document.write( "Answer:  <font color=red>0.07214605</font> (approximate)<BR>\n" );
document.write( "Round it however your teacher instructs.\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "We use the binomial distribution since...<ol><li>The baseball player gets a hit or they do not. There are two outcomes. </li><li>Each trial, or at bat, has the same probability</li><li>Each trial is independent. </li></ol>The three criteria for a binomial have been met.\r<BR>\n" );
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document.write( "The binomial probability formula is<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "n = sample size = number of attempts or at bats = 8<BR>\n" );
document.write( "p = batting average = probability of a hit = 0.216<BR>\n" );
document.write( "x = number of hits<BR>\n" );
document.write( "x spans from x = 0 to x = 8.<BR>\n" );
document.write( "i.e. x is from the set {0,1,2,3,4,5,6,7,8}\r<BR>\n" );
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document.write( "The nCx refers to the nCr combination formula. <BR>\n" );
document.write( "Some textbooks call it the \"choose\" formula. <BR>\n" );
document.write( "These values can be computed with the nCr formula or by use of Pascal's Triangle.\r<BR>\n" );
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document.write( "Let's compute the binomial probability that the player gets x = 0 hits<BR>\n" );
document.write( "B(x) = (nCx)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (8Cx)*(0.216^x)*(1-0.216)^(8-x)<BR>\n" );
document.write( "B(x) = (8Cx)*(0.216^x)*(0.784)^(8-x)<BR>\n" );
document.write( "B(0) = (8C0)*(0.216^0)*(0.784)^(8-0)<BR>\n" );
document.write( "B(0) = 1*(0.216^0)*(0.784)^(8-0)<BR>\n" );
document.write( "B(0) = 0.14273435<BR>\n" );
document.write( "There is roughly a 14.27% chance that the batter doesn't get any hits in the 8 at bats.\r<BR>\n" );
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document.write( "Follow similar steps for x = 1 through x = 3 to find:<BR>\n" );
document.write( "B(1) = 0.31459816<BR>\n" );
document.write( "B(2) = 0.30336251<BR>\n" );
document.write( "B(3) = 0.16715893\r<BR>\n" );
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document.write( "Then adding up those results gives<BR>\n" );
document.write( "B(0) + B(1) + B(2) + B(3) <BR>\n" );
document.write( "= 0.14273435 + 0.31459816 + 0.30336251 + 0.16715893<BR>\n" );
document.write( "= 0.92785395<BR>\n" );
document.write( "This is the probability that the batter gets at most 3 hits, i.e. 3 or less hits.\r<BR>\n" );
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document.write( "Subtract this from 1 to get the final answer.<BR>\n" );
document.write( "1 - 0.92785395 = <font color=red>0.07214605</font>\r<BR>\n" );
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document.write( "Why are we subtracting from 1? <BR>\n" );
document.write( "Because of this scratch work<BR>\n" );
document.write( "( <font color=blue>B(0) + B(1) + B(2) + B(3)</font> ) + ( <font color=red>B(4) + B(5) + B(6) + B(7) + B(8)</font> ) = 1<BR>\n" );
document.write( "which solves to<BR>\n" );
document.write( "<font color=red>B(4) + B(5) + B(6) + B(7) + B(8)</font> = 1 - ( <font color=blue>B(0) + B(1) + B(2) + B(3)</font> )\r<BR>\n" );
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document.write( "Think of it like <BR>\n" );
document.write( "<font color=red> sum of B(4) through B(8) </font> = 1 - ( <font color=blue> sum of B(0) through B(3) </font>)<BR>\n" );
document.write( "or,<BR>\n" );
document.write( "<font color=red> P(at least 4 hits) </font> = 1 - <font color=blue>P(at most 3 hits) </font>\r<BR>\n" );
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document.write( "Or alternatively you can compute values B(4) through B(8) and add them up to get the answer more directly. <BR>\n" );
document.write( "There's not much of a trade-off between both options since we're adding up the same number of terms more or less.\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "A much faster alternative is to use software such as a spreadsheet.<BR>\n" );
document.write( "The <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c\">BinomDist</a> function in Excel and many other spreadsheets is what we want to use.\r<BR>\n" );
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document.write( "The command to type in would be <font color=red>=1-BinomDist(3,8,0.216,1)</font><BR>\n" );
document.write( "The BinomDist(3,8,0.216,1) portion computes the sum from B(0) up to B(3) including both endpoints. Then we subtract from 1 as explained earlier.<BR>\n" );
document.write( "The equal sign up front is to tell the spreadsheet software to compute the formula rather than display as plaintext.<BR>\n" );
document.write( "Refer to the spreadsheet help manual, or the link I posted, for more information how the BinomDist function works.\r<BR>\n" );
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document.write( "If you have access to a TI83 calculator or similar, then check out <a  rel=nofollow HREF=\"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">this article</a><BR>\n" );
document.write( "Make sure you use the CDF and not PDF. \r<BR>\n" );
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document.write( "Various free online binomial distribution calculators can achieve this same goal.<BR>\n" );
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