document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197154>Question 1197154</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The random variable X has a binomial distribution with n = 7 and p = 0.44.\r<BR>\n" );
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document.write( "Calculate P(4).  Enter your answer as a decimal with four decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830311 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830311\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If your teacher doesn't care to see the steps and/or time is very short, then I recommend a calculator or spreadsheet. \r<BR>\n" );
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document.write( "Here are two such examples of a free online calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/binomial-distribution\">https://www.omnicalculator.com/statistics/binomial-distribution</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/binomial-distribution-calculator/\">https://www.statology.org/binomial-distribution-calculator/</a>\r<BR>\n" );
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document.write( "If you want to use a spreadsheet, then the command to type in is <font color=blue>=BINOM.DIST(4,7,0.44, 0)</font><BR>\n" );
document.write( "Alternatively you can use BINOMDIST in place of BINOM.DIST<BR>\n" );
document.write( "Don't forget about the equal sign up front.<BR>\n" );
document.write( "Check out this page for further documentation of how the BINOMDIST function works<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://support.microsoft.com/en-us/office/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c\">https://support.microsoft.com/en-us/office/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c</a><BR>\n" );
document.write( "This documentation page is for excel, but the function works with other spreadsheet programs like OpenOffice and Google Sheets.\r<BR>\n" );
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document.write( "On TI83 and TI84 calculators, press the button labeled \"2nd\" in the top left corner. Then hit the VARS key. Scroll down until you reach binompdf<BR>\n" );
document.write( "and hit enter. Then type in the following<BR>\n" );
document.write( "<font color=blue>binompdf(7,0.44,4)</font><BR>\n" );
document.write( "Unfortunately the order of these input values is slightly different from BINOMDIST mentioned earlier. Be sure to review your calculator's manual to get the correct order of values. \r<BR>\n" );
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document.write( "In GeoGebra, the command to type in is <font color=blue>BinomialDist(7, 0.44, 4, false)</font>\r<BR>\n" );
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document.write( "Feel free to pick your favorite calculator method.\r<BR>\n" );
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document.write( "Using any of those calculators gives an <font color=red>answer of 0.2304</font> when rounding to four decimal places.\r<BR>\n" );
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document.write( "If you don't have access to such calculators, or your teacher won't allow it, then the standard method is to use the Binomial Distribution formula that the tutor @ewatrrr has mentioned. <BR>\n" );
document.write( "That's the method I'd recommend if you are in exam settings. \r<BR>\n" );
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document.write( "But let's look at this a different way so we can see where the binomial formula comes from.\r<BR>\n" );
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document.write( "The term \"binomial\" means there are two outcomes.<BR>\n" );
document.write( "bi = 2\r<BR>\n" );
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document.write( "Think of a coin flip.<BR>\n" );
document.write( "In this case, the coin is biased toward one side.<BR>\n" );
document.write( "Let's say \"heads\" has a probability of 0.44<BR>\n" );
document.write( "That means \"tails\" has probability of 1-0.44 = 0.56\r<BR>\n" );
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document.write( "We'll flip the coin n = 7 times.<BR>\n" );
document.write( "We could flip a single coin 7 times, or we could flip 7 coins exactly once each. <BR>\n" );
document.write( "The two situations are identical.\r<BR>\n" );
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document.write( "P(4) represents the \"The probability of exactly 4 heads showing up out of 7 coin tosses\".<BR>\n" );
document.write( "The notation is equivalent to P(x = 4).\r<BR>\n" );
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document.write( "Assuming each coin toss is independent, we multiply out the probabilities<BR>\n" );
document.write( "Let's start with the heads<BR>\n" );
document.write( "(0.44)(0.44)(0.44)(0.44) = (0.44)^4\r<BR>\n" );
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document.write( "Then the tails<BR>\n" );
document.write( "(0.56)(0.56)(0.56) = (0.56)^3\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "probability of four heads = (0.44)^4<BR>\n" );
document.write( "probability of three tails = (0.56)^3\r<BR>\n" );
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document.write( "Therefore (0.44)^4*(0.56)^3 represents the probability of getting the sequence HHHH TTT\r<BR>\n" );
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document.write( "The expression (0.44)^4*(0.56)^3 applies to that sequence only. <BR>\n" );
document.write( "But we could arrange the heads and tails.\r<BR>\n" );
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document.write( "For instance, we could go from this<BR>\n" );
document.write( "HHHH  TTT<BR>\n" );
document.write( "to this<BR>\n" );
document.write( "HHHT  HTT\r<BR>\n" );
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document.write( "The question is: how can we count the number of ways to arrange these heads and tails?\r<BR>\n" );
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document.write( "That's where the nCr combination formula comes in.<BR>\n" );
document.write( "Some textbooks will call this the \"choose\" formula.\r<BR>\n" );
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document.write( "Imagine slips of paper labeled 1 through 7. Then place those pieces of paper into a hat.<BR>\n" );
document.write( "We'll randomly select 4 slips of paper.\r<BR>\n" );
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document.write( "One possible sequence is 1,2,5,7<BR>\n" );
document.write( "Those numbers will then tell us where to place the \"H\"s to get this sequence of coin flips: H,H,_,_,H,_,H<BR>\n" );
document.write( "I've filled slots 1,2,5 and 7.\r<BR>\n" );
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document.write( "Each black space will then get a T to get HHTT  HTH\r<BR>\n" );
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document.write( "The process of selecting those 4 pieces of paper can be counted by using the nCr combination formula.\r<BR>\n" );
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document.write( "We have n = 7 pieces of paper and r = 4 selections<BR>\n" );
document.write( "Let's compute the nCr value<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "7 C 4 = (7!)/(4!*(7-4)!)<BR>\n" );
document.write( "7 C 4 = (7!)/(4!*3!)<BR>\n" );
document.write( "7 C 4 = (7*6*5*4!)/(4!*3!)<BR>\n" );
document.write( "7 C 4 = (7*6*5)/(3!)<BR>\n" );
document.write( "7 C 4 = (7*6*5)/(3*2*1)<BR>\n" );
document.write( "7 C 4 = (210)/(6)<BR>\n" );
document.write( "7 C 4 = 35<BR>\n" );
document.write( "Put another way, we have 7*6*5 = 210 permutations, and then we divide by 3*2*1 = 6 to adjust for the fact that order doesn't matter.\r<BR>\n" );
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document.write( "So that's where the 35 comes from in the equation @ewatrrr mentioned.\r<BR>\n" );
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document.write( "You can also use Pascal's Triangle to determine this. Locate the row that has 1,7,... at the start. Then count 4+1 = 5 spaces to arrive at 35.\r<BR>\n" );
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document.write( "Recall that (0.44)^4*(0.56)^3 represented the probability of getting the sequence HHHH  TTT.\r<BR>\n" );
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document.write( "Then we counted 35 ways to arrange the four H's and three T's.\r<BR>\n" );
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document.write( "This means we'll stick 35 out front to get <BR>\n" );
document.write( "35*(0.44)^4*(0.56)^3\r<BR>\n" );
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document.write( "Computing that expression will get roughly 0.2303789694976 which then rounds to <font color=red>0.2304</font> which we got earlier.<BR>\n" );
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