document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172247>Question 1172247</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear tutor, Please help me solving these problems\r<BR>\n" );
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document.write( "1)	An unbiased die is tossed. Find the probability that the number obtained is:<BR>\n" );
document.write( "(a) An event number 	(b) Less than 4	(c) Greater than 6\r<BR>\n" );
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document.write( "2)	A card is randomly drawn from a standard deck of 52 playing cards. Find the probability that the card drawn is:	<BR>\n" );
document.write( "(a) A king		(b) A diamond       (c) Not a diamond      	(d) A king of diamonds\r<BR>\n" );
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document.write( "3)	A letter is randomly chosen from the word NATIONAL. What is the probability that it will be?<BR>\n" );
document.write( "(a) The letter A		(b) A vowel		(c) The letter S\r<BR>\n" );
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document.write( "4)	A box contains 3 red discs, 4 green discs, and 5 blue discs. One disc is randomly drawn from the box. What is probability that will be:<BR>\n" );
document.write( "(a) red disc		(b) A green disc	(c) A red disc or green disc	(d) Not a blue disc<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #797207 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=797207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3>You should only post one question at a time. I'll do the first 2 problems to get you started.\r<BR>\n" );
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document.write( "Problem 1\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "There are 3 even numbers (2,4,6) out of 6 total (1 through 6). <BR>\n" );
document.write( "So 3/6 = 1/2 is the probability of rolling an even number.<BR>\n" );
document.write( "Or you could say \"half of the values are even\".\r<BR>\n" );
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document.write( "<font color=red>Answer: 1/2</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "We could use the same idea as part (a) above, but let's take a more formal approach. This approach is likely what many math textbooks use.\r<BR>\n" );
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document.write( "We'll use the concept of event space and sample space.<BR>\n" );
document.write( "The event space is the set of all desired outcomes.<BR>\n" );
document.write( "The sample space is the set of all possible outcomes (desired or not).<BR>\n" );
document.write( "The event space is a subset of the sample space, which means that anything in set E is also in set S, but not vice versa.\r<BR>\n" );
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document.write( "Let's say <BR>\n" );
document.write( "E = event space<BR>\n" );
document.write( "S = sample space<BR>\n" );
document.write( "both represent sets\r<BR>\n" );
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document.write( "Let n(E) represent the number of items in the event space.<BR>\n" );
document.write( "Let n(S) represent the number of items in the sample space.\r<BR>\n" );
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document.write( "Dividing the numbers n(E) and n(S) leads to the probability of selecting an item randomly from the event space.<BR>\n" );
document.write( "We can say<BR>\n" );
document.write( "P(E) = n(E)/n(S)\r<BR>\n" );
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document.write( "With that bit of info in mind, we can tackle the problem.\r<BR>\n" );
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document.write( "E = {1,2,3} = set of values we want, everything less than 4<BR>\n" );
document.write( "S = {1,2,3,4,5,6} = set of all possible values (some values in here are what we don't want)<BR>\n" );
document.write( "everything in E is also in S, but not vice versa\r<BR>\n" );
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document.write( "n(E) = 3 = there are 3 items in set E<BR>\n" );
document.write( "n(S) = 6 = there are 6 items in set S<BR>\n" );
document.write( "P(E) = n(E)/n(S)<BR>\n" );
document.write( "P(E) = 3/6<BR>\n" );
document.write( "P(E) = 1/2<BR>\n" );
document.write( "This approach may seem more tedious, but it's handy to get a formal structure of how probability is set up.\r<BR>\n" );
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document.write( "<font color=red>Answer: 1/2</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Event space = {  } = empty set<BR>\n" );
document.write( "Sample space = {1,2,3,4,5,6}<BR>\n" );
document.write( "The event space is empty because there are no values greater than 6 on a standard die. This leads to a probability of 0 which says \"the probability of getting a value greater than 6 is impossible\".\r<BR>\n" );
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document.write( "More formally,<BR>\n" );
document.write( "P(E) = n(E)/n(S)<BR>\n" );
document.write( "P(E) = 0/6<BR>\n" );
document.write( "P(E) = 0\r<BR>\n" );
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document.write( "<font color=red>Answer: 0</font>\r<BR>\n" );
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document.write( "Problem 2\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Event space = {King of Hearts, King of Diamonds, King of Spades, King of Clubs}<BR>\n" );
document.write( "Sample space = {52 cards in a standard deck}<BR>\n" );
document.write( "We have 4 items in the event space, out of 52 items in the sample space. <BR>\n" );
document.write( "We get 4/52 = 1/13.\r<BR>\n" );
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document.write( "<font color=red>Answer: 1/13</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "There are 13 diamond cards out of 52 total. <BR>\n" );
document.write( "We get the probability 13/52 = 1/4<BR>\n" );
document.write( "You could also look at it like \"there is one diamond suit out of 4 suits total\".\r<BR>\n" );
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document.write( "<font color=red>Answer: 1/4</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "We'll use the result from part (b). Subtract it from 1<BR>\n" );
document.write( "This works because both events \"getting diamonds\" and \"getting not diamonds\" are complementary. <BR>\n" );
document.write( "You either get a diamond or you don't.<BR>\n" );
document.write( "P(not diamonds) = 1 - P(diamonds)<BR>\n" );
document.write( "P(not diamonds) = 1 - 1/4<BR>\n" );
document.write( "P(not diamonds) = 3/4<BR>\n" );
document.write( "We can confirm this by noting there are 3 suits that aren't diamond out of 4 suits total.\r<BR>\n" );
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document.write( "<font color=red>Answer: 3/4</font>\r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "There's one card that is a king of diamonds out of 52 total.\r<BR>\n" );
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document.write( "<font color=red>Answer: 1/52</font></font> </SPAN>\n" );
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