document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191369>Question 1191369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pair of 6 side dice is thrown. A wins if the sum is 5 and the game stops immediately. B wins if the sum is 7 and the game stops immediately. What is the probability of A winning?\r<BR>\n" );
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document.write( "I know that I'm supposed to solve this using geometric series but can't seem to find clear examples of this kind of question. I tried 1/(1+(1-(4/36)) and my answer was 9/17 which is aprox 0.52 but that seems off to me. </SPAN>\n" );
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document.write( "(1) Since the probability of getting a 5 is less than the probability of getting a 7, in the long run the probability of A winning should be less than 0.5 -- so your answer should be off.  Note the formula you are using to get your answer is not correct....<br><BR>\n" );
document.write( "(2) The probability of A winning depends on whether A or B goes first.  Your statement of the problem doesn't specify....<br><BR>\n" );
document.write( "P(5 on any throw) = 4/36 = 1/9<BR>\n" );
document.write( "P(not 5 on any throw) = 8/9<BR>\n" );
document.write( "P(7 on any throw) = 6/36 = 1/6<BR>\n" );
document.write( "P(not 7 on any throw) = 5/6<br><BR>\n" );
document.write( "If A goes first....<br><BR>\n" );
document.write( "A wins if the sequence of throws is<BR>\n" );
document.write( "A 5, or<BR>\n" );
document.write( "A not 5, B not 7, A 5, or<BR>\n" );
document.write( "A not 5, B not 7, A not 5, B not 7, A 5, or<BR>\n" );
document.write( "...<BR>\n" );
document.write( "The probability of A winning if A goes first is then<br><BR>\n" );
document.write( "(1/9)+(8/9)(5/6)(1/9)+(8/9)(5/6)(8/9)(5/6)(1/9)+...<br><BR>\n" );
document.write( "Using the formula for the infinite sum of a geometric series (first term 1/9; common ratio (8/9)(5/6)=20/27)....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F9%29%2F%281-%288%2F9%29%285%2F6%29%29+=+%281%2F9%29%2F%281-20%2F27%29+=+%281%2F9%29%2F%287%2F27%29+=+%281%2F9%29%2827%2F7%29+=+3%2F7 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then it should be true that if A goes first the probability of B winning is 4/7.  Let's verify that using the same kind of calculations as above.<br><BR>\n" );
document.write( "If A goes first....<br><BR>\n" );
document.write( "B wins if the sequence of throws is<BR>\n" );
document.write( "A not 5, B 7, or<BR>\n" );
document.write( "A not 5, B not 7, A not 5, B 7, or<BR>\n" );
document.write( "A not 5, B not 7, A not 5, B not 7, A not 5, B 7, or<BR>\n" );
document.write( "...<BR>\n" );
document.write( "The probability of A winning if A goes first is then<br><BR>\n" );
document.write( "(8/9)(1/6)+(8/9)(5/6)(8/9)(1/6)+(8/9)(5/6)(8/9)(5/6)(8/9)(1/6)+...<br><BR>\n" );
document.write( "Using the formula for the infinite sum of a geometric series....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%284%2F27%29%2F%281-%288%2F9%29%285%2F6%29%29+=+%284%2F27%29%2F%281-20%2F27%29+=+%284%2F27%29%2F%287%2F27%29+=+%284%2F27%29%2827%2F7%29+=+4%2F7 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now here is a somewhat advanced concept regarding this problem.<br><BR>\n" );
document.write( "Since the common ratio in both sequences is the same -- representing each of A and B rolling the dice and NOT getting the number they want -- the ratio of A winning to B winning is the same as the ratio of A winning on his first throw to B winning on his first throw.<br><BR>\n" );
document.write( "P(A wins on first throw) = 1/9<BR>\n" );
document.write( "P(B wins on his first throw) = (8/9)(1/6) = 8/54 = 4/27<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F9%29%2F%284%2F27%29+=+%281%2F9%29%2827%2F4%29=3%2F4 ALIGN=MIDDLE  ALT=\"%281%2F9%29%2F%284%2F27%29+=+%281%2F9%29%2827%2F4%29=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then since the ratio of A winning to B winning is 3:4, the probability that A wins is 3/7 and the probability that B wins is 4/7.<br><BR>\n" );
document.write( "If B goes first....<br><BR>\n" );
document.write( "Again, the statement of the problem did not specify which player goes first, so let's look at the problem again if B goes first.<br><BR>\n" );
document.write( "Using the same argument as in the preceding paragraph, the ratio of the probabilities of B or A winning is the ratio of the probabilities that each player wins on his first throw.<br><BR>\n" );
document.write( "P(B goes first and rolls 7 on first throw) = 1/6<BR>\n" );
document.write( "P(B goes first, does not roll 7, and A rolls 5 on first throw) = (5/6)(1/9) = 5/54<br><BR>\n" );
document.write( "The ratio of B winning to A winning is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F6%29%2F%285%2F54%29+=+%281%2F6%29%2854%2F5%29+=+9%2F5 ALIGN=MIDDLE  ALT=\"%281%2F6%29%2F%285%2F54%29+=+%281%2F6%29%2854%2F5%29+=+9%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this case (B rolling first), the ratio of B winning to A winning is 9:5, so the probability of B winning is 9/14 and the probability of A winning is 5/14.<br><BR>\n" );
document.write( "You should find those probabilities if you analyze the case of B going first using infinite geometric series, as done in the first part of this response for the case of A going first.<br><BR>\n" );
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