document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022199>Question 1022199</A>: <I><SPAN STYLE=\"word-break: break-all;\"> About 26% of ice-cream sales are vanilla. 14% account for chocolate sales.  Suppose 184 customers  go to a grocery store.<BR>\n" );
document.write( "a)what is the probably that 50 or more will buy vanilla?<BR>\n" );
document.write( "b) what is the probability 12 or more will buy chocolate<BR>\n" );
document.write( "c) What is the probability that someone who is buying ice cream will buy chocolate or vanilla?<BR>\n" );
document.write( "d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream?\r<BR>\n" );
document.write( "\n" );
document.write( "I'm so stuck on how to set up the problem. <BR>\n" );
document.write( "Elin </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #637988 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=637988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "About 26% of ice-cream sales are vanilla. 14% account for chocolate sales.  Suppose 184 customers  go to a grocery store.<BR>\n" );
document.write( "a)what is the probably that 50 or more will buy vanilla?<BR>\n" );
document.write( "b) what is the probability 12 or more will buy chocolate<BR>\n" );
document.write( "c) What is the probability that someone who is buying ice cream will buy chocolate or vanilla?<BR>\n" );
document.write( "d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Setting up and example of solution:<BR>\n" );
document.write( "Here we have a case where:<BR>\n" );
document.write( "1. The number of trials (n=184) is known.<BR>\n" );
document.write( "2. Each trial is a Bernoulli experiment (true or false, e.g. vanilla or not vanilla, chocolate, or not cholate).<BR>\n" );
document.write( "3. All trials are (assumed) random and independent of each other.<BR>\n" );
document.write( "4. The probability of each outcome is constant throughout the experiment (0.26 for vanilla, 0.14 for chocolate, meaning 0.40 for either one of the two).<BR>\n" );
document.write( "Under these circumstances, we can model the situation with a binomial distribution, which says that the probability of k successes out of n trials each with a probability of success p is given by:<BR>\n" );
document.write( "P(k;n;p)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Ck%29%2Ap%5Ek%2A%281-p%29%5E%28n-k%29 ALIGN=MIDDLE  ALT=\"C%28n%2Ck%29%2Ap%5Ek%2A%281-p%29%5E%28n-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.............(1)<BR>\n" );
document.write( "where <BR>\n" );
document.write( "C(n,k)=n!/(k!(n-k)!) is coefficient of combination of k objects out of n.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a)<BR>\n" );
document.write( "So we need the probability of k>=50 customers would buy vanilla.<BR>\n" );
document.write( "We need <BR>\n" );
document.write( "P(k>=50;n;p)<BR>\n" );
document.write( "=sum of all cases of P(k;n;p) for k=50,51,....184.<BR>\n" );
document.write( "OR<BR>\n" );
document.write( "= 1-sum of all cases of P(k;n;p) for k=0,1,2,3....49.<BR>\n" );
document.write( " <BR>\n" );
document.write( "This means a lot of calculations using equation (1).<BR>\n" );
document.write( "You would probably need a scientific calculator with the binomial distribution built-in, or an application such as R.<BR>\n" );
document.write( "The latter gives<BR>\n" );
document.write( "P(k>=50;184;0.26)=0.3854 [pbinom(49,184,0.26]<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you don't have an advanced scientific calculator, you could use, in this particular case the normal approximation to the binomial, which uses<BR>\n" );
document.write( "mean=&mu;=np=184*0.26=47.84,<BR>\n" );
document.write( "standard deviation=&sigma;=sqrt(variance)=sqrt(np(1-p))=sqrt(184*.26*.74)=5.95<BR>\n" );
document.write( "The required range is from 50 to 184, but since 49.5+ would round to 50, so we will calculate the range of 49.5<=k<=184 (called the continuity correction).<BR>\n" );
document.write( "Now we need to convert k to Z (for normal distribution), <BR>\n" );
document.write( "Zmin=(Xmin-&mu;)/&sigma;=(49.5-47.84)/5.95=0.27899<BR>\n" );
document.write( "We then look up the upper tail from a normal distribution table to get the probability of Z>=0.27900 (upper tail) to be 0.3901, not that far from the exact answer of 0.3854 (1.2% over).<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can proceed with the other parts similarly.<BR>\n" );
document.write( "Feel free to post your answer for checking. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );