document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198328>Question 1198328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given an unfair coin with probability of obtaining a head equal to<BR>\n" );
document.write( "1/3, 000, 000, 000. You toss this coin 6,000,000,000 times. Let A be the<BR>\n" );
document.write( "event that you get “tails for all the tosses”. Let B be the event that you get<BR>\n" );
document.write( "“heads for all the tosses”.<BR>\n" );
document.write( "i) Approximate P (A).<BR>\n" );
document.write( "ii) Approximate P (A ∪ B) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831933 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+ ALIGN=MIDDLE  ALT=\"N+\"   DISABLED_style= \"max-width:90%; height:auto;\" >be the number of heads among the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > tosses. Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∼ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+p ALIGN=MIDDLE  ALT=\"+p\"   DISABLED_style= \"max-width:90%; height:auto;\" >). \r<BR>\n" );
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document.write( "given:\r<BR>\n" );
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document.write( "probability of obtaining a head equal to:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3000000000=1%2F%283+%2A10%5E9%29 ALIGN=MIDDLE  ALT=\"1%2F3000000000=1%2F%283+%2A10%5E9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "You toss this coin <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6000000000=+6%2A10%5E9 ALIGN=MIDDLE  ALT=\"6000000000=+6%2A10%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" > times\r<BR>\n" );
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document.write( "Here, we have small <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+1%2F%283+%2A10%5E9%29 ALIGN=MIDDLE  ALT=\"p+=+1%2F%283+%2A10%5E9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and large <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+6%2A10%5E9 ALIGN=MIDDLE  ALT=\"n+=+6%2A10%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, we can apply Poisson approximation.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+p ALIGN=MIDDLE  ALT=\"+p\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is well-approximated by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28alpha%29 ALIGN=MIDDLE  ALT=\"P%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha+=+np+=+2 ALIGN=MIDDLE  ALT=\"alpha+=+np+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(a) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29 ALIGN=MIDDLE  ALT=\"P%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+0 ALIGN=MIDDLE  ALT=\"N+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >] = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E-2%282%5E0%2F0%21%29=1%2Fe%5E2 ALIGN=MIDDLE  ALT=\"e%5E-2%282%5E0%2F0%21%29=1%2Fe%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >≈ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.1353 ALIGN=MIDDLE  ALT=\"0.1353\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(b) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∪ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+0 ALIGN=MIDDLE  ALT=\"N+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >]+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+n ALIGN=MIDDLE  ALT=\"N+=+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >] = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E-2%282%5E0%2F0%21%29%2Be%5E-2%28%282%5E6%2A10%5E9%29%2F%286%2A10%5E9%29%21%29 ALIGN=MIDDLE  ALT=\"e%5E-2%282%5E0%2F0%21%29%2Be%5E-2%28%282%5E6%2A10%5E9%29%2F%286%2A10%5E9%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " The second term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=extremely ALIGN=MIDDLE  ALT=\"extremely\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+small ALIGN=MIDDLE  ALT=\"+small\"   DISABLED_style= \"max-width:90%; height:auto;\" > compared to the first one. \r<BR>\n" );
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document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > ∪ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is approximately the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28A%29 ALIGN=MIDDLE  ALT=\"P%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );