document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=488990>Question 488990</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If I drop three coins on a table, what is the probability of finding<BR>\n" );
document.write( "   1. exactly 2 heads<BR>\n" );
document.write( "   2. at most 1 head<BR>\n" );
document.write( "   3. at least 2 heads<BR>\n" );
document.write( "   4. at least 1 head<BR>\n" );
document.write( "sample space = 8\r<BR>\n" );
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document.write( "I came up with HHT, HTH, THH,HTT,THT,TTH,TTT,HHH.<BR>\n" );
document.write( "state the probability of each in the form of fraction and percent.<BR>\n" );
document.write( "For 1,2,3 I came up with 1/4 or .25, and for 4 = 1/6.<BR>\n" );
document.write( "Have I done this correctly, or can you show me a better way to show this??<BR>\n" );
document.write( "Is this a form of \"classical probability\"??<BR>\n" );
document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #333504 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=333504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> p(heads) = .5<BR>\n" );
document.write( "p(tails) = .5<BR>\n" );
document.write( "since the probability of success and failure are the same, then the probability for each occurrence is the same.<BR>\n" );
document.write( "that probability is .5*.5*.5 = .125<BR>\n" );
document.write( "what differs is the ways in which each occurrence can happen.<BR>\n" );
document.write( "3 heads can only happen in 1 way.<BR>\n" );
document.write( "3 tails can only happen in 1 way.<BR>\n" );
document.write( "1 head and 2 tails can happen in 3 ways.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "htt<BR>\n" );
document.write( "tht<BR>\n" );
document.write( "tth<BR>\n" );
document.write( "2 heads and 1 tail can happen in 3 ways.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "hht<BR>\n" );
document.write( "hth<BR>\n" );
document.write( "thh<BR>\n" );
document.write( "the formula for determining the ways they can happen is the combination formula.<BR>\n" );
document.write( "that formula is<BR>\n" );
document.write( "nCx = n! / (x! * (n-x)!))<BR>\n" );
document.write( "you are drawing 3 coins and you want exactly 1 head.<BR>\n" );
document.write( "the formula for that becomes:<BR>\n" );
document.write( "n = 3<BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "nCx = 3! / (1! * 2!) = 3! / 2! = (3 * 2!) / 2! = 3<BR>\n" );
document.write( "3! means 3 * 2 * 1<BR>\n" );
document.write( "2! means 2 * 1<BR>\n" );
document.write( "1! means 1<BR>\n" );
document.write( "the total probability of all possible occurrences has to equal to 1.<BR>\n" );
document.write( "for your 3 coins, here's how the probabilities would be calculated.<BR>\n" );
document.write( "probability of 0 heads would be:<BR>\n" );
document.write( "3C0 * (.5)^3 = 3! / (0! * 3!) * .125 = 3! / 3! * .125 = 1 * .125 = .125<BR>\n" );
document.write( "0! is always equal to 1<BR>\n" );
document.write( "1! is always equal to 1<BR>\n" );
document.write( "probability of 1 head would be:<BR>\n" );
document.write( "3C1 * (.5)^3 = 3! / (1! * 2!) * .125 = 3! / 2! * .125 = 3 * .125 = .375<BR>\n" );
document.write( "probability of 2 heads would be:<BR>\n" );
document.write( "3C2 * (.5)^2 = 3! / (2! * 1!) * .125 = 3! / 2! * .125 = 3 * .125 = .375<BR>\n" );
document.write( "probability of 3 heads would be:<BR>\n" );
document.write( "3C3 * (.5)^2 = 3! / (3! * 0!) * .125 = 3! / 3! * .125 = 1 * .125 = .125 <BR>\n" );
document.write( "your total probability is:<BR>\n" );
document.write( ".125 + .125 + .375 + .375 = .250 + .750 = 1.0<BR>\n" );
document.write( "this is what it should be so we calculated correctly.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "the general formula is:<BR>\n" );
document.write( "nCx * p^x * q^(n-x)<BR>\n" );
document.write( "p is the probability of success<BR>\n" );
document.write( "q is the probability of failure.\r<BR>\n" );
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document.write( "in your experiment, the probability of success and the probability of failure were the same so it didn't matter.\r<BR>\n" );
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document.write( "in other case, they will be different.\r<BR>\n" );
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document.write( "suppose the coin was weighted so the probability of heads was .7 and the probability of tails was .3\r<BR>\n" );
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document.write( "the same problem would give you the following probabilities.<BR>\n" );
document.write( "probability of exactly 0 heads = 1 * .7^0 * .3^3 = .027<BR>\n" );
document.write( "probability of exactly 1 head = 3 * .7^1 * .3^2 = .189<BR>\n" );
document.write( "probability of exactly 2 heads = 3 * .7^2 * .3^1 = .441<BR>\n" );
document.write( "probability of exactly 3 heads = 1 * .7^3 = .343\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "the total probability is still equal to 1.<BR>\n" );
document.write( "the number of ways each occurrence can happen is the same.<BR>\n" );
document.write( "the probability of each occurrence, however, is different.\r<BR>\n" );
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document.write( "the answer to your questions from the data for equal probabilities is:\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "If I drop three coins on a table, what is the probability of finding<BR>\n" );
document.write( "1. exactly 2 heads<BR>\n" );
document.write( "exactly 2 heads would be .375<BR>\n" );
document.write( "2. at most 1 head<BR>\n" );
document.write( "at most 1 head would be exactly 0 and exactly 1 which would be .125 + .375 = .5<BR>\n" );
document.write( "3. at least 2 heads<BR>\n" );
document.write( "at least 2 heads would be exactly 2 and exactly 3 which would be .375 and .125 = .5<BR>\n" );
document.write( "4. at least 1 head<BR>\n" );
document.write( "at least 1 head would be exactly 1 and exactly 2 and exactly 3 which would be .375 + .375 + .125 = .875<BR>\n" );
document.write( "not that this is also 1 minus exactly 0 which would be equal to 1 - .125 = .875.\r<BR>\n" );
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document.write( ".125 = 1/8<BR>\n" );
document.write( ".375 = 3/8\r<BR>\n" );
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document.write( "i think you tallied the possibilities correctly but may not have tallied the probabilities correctly.\r<BR>\n" );
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document.write( "doing it the way you did it, i got:<BR>\n" );
document.write( "HHH<BR>\n" );
document.write( "HHT<BR>\n" );
document.write( "HTH<BR>\n" );
document.write( "THH<BR>\n" );
document.write( "HTT<BR>\n" );
document.write( "THT<BR>\n" );
document.write( "TTH<BR>\n" );
document.write( "TTT<BR>\n" );
document.write( "this part seems to be ok.<BR>\n" );
document.write( "looking at this set of data, however, i get the following.<BR>\n" );
document.write( "probability of exactly 0 heads is 1/8 (TTT)<BR>\n" );
document.write( "probability of exactly 1 head is 3/8 (HTT, THT, TTH)<BR>\n" );
document.write( "probability of exactly 2 heads is 3/8 (HHT, HTH, THH)<BR>\n" );
document.write( "probability of exactly 3 heads is 1/8 (HHH)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the answers to the questions are:<BR>\n" );
document.write( "1. exactly 2 heads = 3/8<BR>\n" );
document.write( "2. at most 1 head = 4/8<BR>\n" );
document.write( "3. at least 2 heads = 4/8<BR>\n" );
document.write( "4. at least 1 head = 7/8\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );