document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190165>Question 1190165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, please help me out! I am a bit confused.\r<BR>\n" );
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document.write( "A bag contains 8 blue, 7 red, 5 pink and 10 orange jelly beans. Six jellybeans are to be selected at random without replacement.\r<BR>\n" );
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document.write( "A. What type of distribution is this? (Uniform, Binomial, Geometric, or Hypergeometric)\r<BR>\n" );
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document.write( "B. What is the probability that at least one of the jelly beans picked is orange?\r<BR>\n" );
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document.write( "C. What is the expected number of pink jelly beans to be selected? (Which expectance formula should I use? They all vary for the type of distribution so I am confused) </SPAN>\n" );
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document.write( "Answers:<ul><li>A) <font color=red>Hypergeometric</font></li><li>B) <font color=red>0.93472275 approximately</font></li><li>C) <font color=red>1</font></li></ul>==================================================<BR>\n" );
document.write( "Explanation for part A\r<BR>\n" );
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document.write( "The uniform distribution is where the probability of any outcome is the same. For example, rolling a single six-sided die has each side with uniform probability 1/6. <BR>\n" );
document.write( "We rule out \"uniform distribution\" because the probabilities aren't all the same for this jelly bean problem.\r<BR>\n" );
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document.write( "The binomial distribution can be ruled out as well. This is due to the phrasing \"without replacement\". \r<BR>\n" );
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document.write( "The geometric distribution isn't used because we aren't interested in questions like \"what is the probability of getting blue on the first selection? second selection? etc\".\r<BR>\n" );
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document.write( "We use the hypergeometric distribution, which is effectively the binomial distribution but without replacement. <BR>\n" );
document.write( "This distribution is useful to list out the possible X values and their corresponding P(X) values. <BR>\n" );
document.write( "The X values refer to the counts of a certain color. Refer to parts B or C for more info.\r<BR>\n" );
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document.write( "Explanation for part B\r<BR>\n" );
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document.write( "Let's compute the probability of getting all six jelly beans that aren't orange.<BR>\n" );
document.write( "We have 8 blue + 7 red + 5 pink = 20 beans that aren't orange out of 8+7+5+10 = 30 jelly beans total.\r<BR>\n" );
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document.write( "The probability of getting six beans in a row that aren't orange is<BR>\n" );
document.write( "P(6 not orange) = (20/30)*(19/29)*(18/28)*(17/27)*(16/26)*(15/25)<BR>\n" );
document.write( "P(6 not orange) = 0.06527725<BR>\n" );
document.write( "Notice the numerators counting down (20,19,...) and the denominators are counting down as well (30,29,...). This is caused by the fact the previous selection is not replaced.\r<BR>\n" );
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document.write( "Subtract this value from 1 to get<BR>\n" );
document.write( "P(at least one orange) = 1 - P(6 non orange)<BR>\n" );
document.write( "P(at least one orange) = 1 - 0.06527725<BR>\n" );
document.write( "P(at least one orange) = 0.93472275\r<BR>\n" );
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document.write( "The events \"selecting 6 non orange\" and \"at least one orange\" are complementary events. One or the other must happen. <BR>\n" );
document.write( "Therefore the two event probabilities add to 1, which is why that previous formula is useful.\r<BR>\n" );
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document.write( "Side note: as an alternative path, you can use the formula mentioned in part C. Plug in N = 30, n = 6, K = 20, k = 6 to get the result 0.06527725 which you subtract from 1 to get the final answer for this part.\r<BR>\n" );
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document.write( "Explanation for part C\r<BR>\n" );
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document.write( "X = number of pink selected<BR>\n" );
document.write( "The set of possible X values is {0,1,2,3,4,5}\r<BR>\n" );
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document.write( "Use the hypergeometric probability formula to calculate each P(X) for the set of X values mentioned.\r<BR>\n" );
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document.write( "We'll think of each outcome as \"pink\" vs \"not pink\".<BR>\n" );
document.write( "There are 5 pink and 25 non pink.<ul><li>N = population size = 30 beans total</li><li>n = sample size = 6 beans selected</li><li>K = number of pink overall in population = 5</li><li>k = number of pink selected = varies from 0 to 5 i.e. those X values mentioned earlier</li></ul>The convention is that the uppercase letters N and K go with the population counts (total beans and pink beans); while the lowercase counterparts count the sample values.\r<BR>\n" );
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document.write( "I'll show the steps on how to calculate P(X = 0)<BR>\n" );
document.write( "In other words when k = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(k) = \frac{C(K,k)*C(N-K,n-k)}{C(N,n)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(0) = \frac{C(5,0)*C(30-5,6-0)}{C(30,6)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(0) = \frac{C(5,0)*C(25,6)}{C(30,6)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(0) = \frac{1*177100}{593775}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(0) = 0.298261\"> approximately\r<BR>\n" );
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document.write( "For more information, check out this link about the hypergeometric distribution<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://online.stat.psu.edu/stat414/lesson/7/7.4\">https://online.stat.psu.edu/stat414/lesson/7/7.4</a><BR>\n" );
document.write( "The notation C(n,r) refers to the nCr combination formula. Sometimes it's written in the form of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {n\choose r}\">\r<BR>\n" );
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document.write( "This is a useful free calculator to help check your answers<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stattrek.com/online-calculator/hypergeometric.aspx\">https://stattrek.com/online-calculator/hypergeometric.aspx</a>\r<BR>\n" );
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document.write( "You'll follow similar steps for k = 1 through k = 5\r<BR>\n" );
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document.write( "Here's a table of the results<BR>\n" );
document.write( "The P(X) values are approximate and rounded to 6 decimal places.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >0</TD><TD >0.298261</TD></tr><tr><TD >1</TD><TD >0.447392</TD></tr><tr><TD >2</TD><TD >0.213044</TD></tr><tr><TD >3</TD><TD >0.038735</TD></tr><tr><TD >4</TD><TD >0.002526</TD></tr><tr><TD >5</TD><TD >0.000042</TD></tr>\n" );
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document.write( "Next, we'll multiply each X and P(X) value to form a new column<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >0</TD><TD >0.298261</TD><TD >0</TD></tr><tr><TD >1</TD><TD >0.447392</TD><TD >0.447392</TD></tr><tr><TD >2</TD><TD >0.213044</TD><TD >0.426088</TD></tr><tr><TD >3</TD><TD >0.038735</TD><TD >0.116205</TD></tr><tr><TD >4</TD><TD >0.002526</TD><TD >0.010104</TD></tr><tr><TD >5</TD><TD >0.000042</TD><TD >0.000210</TD></tr>\n" );
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document.write( "From here you need to add up the X*P(X) results<BR>\n" );
document.write( "0+0.447392+0.426088+0.116205+0.010104+0.000210 = 0.999999\r<BR>\n" );
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document.write( "That effectively rounds to 1<BR>\n" );
document.write( "I suspect that the true exact answer is 1 if there wasn't rounding error (when computing the approximate P(X) values).\r<BR>\n" );
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document.write( "We expect about 1 pink jelly bean will be in the sample size of 6.<BR>\n" );
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