document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201351>Question 1201351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A high school baseball player has a 0.153 batting average. In one game, he gets 6 at bats. What is the probability he will get at least 2 hits in the game? </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>0.230583</font><BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "n = 6 = sample size = number of at bats<BR>\n" );
document.write( "p = 0.153 = batting average = probability of success of any at bat\r<BR>\n" );
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document.write( "x = number of successful attempts = number of hits<BR>\n" );
document.write( "x is some integer in the set {0, 1, 2, 3, 4, 5, 6}<BR>\n" );
document.write( "x is between 0 and 6, including each endpoint.\r<BR>\n" );
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document.write( "B(x) = binomial probability<BR>\n" );
document.write( "B(x) = (n C x)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (6 C x)*(0.153^x)*(1-0.153)^(6-x)<BR>\n" );
document.write( "B(x) = (6 C x)*(0.153^x)*(0.847)^(6-x)<BR>\n" );
document.write( "The nCx refers to the nCr combination formula.\r<BR>\n" );
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document.write( "\"at least 2 hits\" translates to \"2 or more hits\"<BR>\n" );
document.write( "We want to find the probability that x = 2 or x > 2<BR>\n" );
document.write( "We need to calculate B(2)+B(3)+B(4)+B(5)+B(6)<BR>\n" );
document.write( "A shortcut is to calculate B(0)+B(1) first and then subtract from 1.\r<BR>\n" );
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document.write( "Plug in x = 0<BR>\n" );
document.write( "B(x) = (6 C x)*(0.153^x)*(0.847)^(6-x)<BR>\n" );
document.write( "B(0) = (6 C 0)*(0.153^0)*(0.847)^(6-0)<BR>\n" );
document.write( "B(0) = 1*(0.153^0)*(0.847)^6<BR>\n" );
document.write( "B(0) = 0.36923296009284<BR>\n" );
document.write( "B(0) = 0.369233\r<BR>\n" );
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document.write( "Plug in x = 1<BR>\n" );
document.write( "B(x) = (6 C x)*(0.153^x)*(0.847)^(6-x)<BR>\n" );
document.write( "B(1) = (6 C 1)*(0.153^1)*(0.847)^(6-1)<BR>\n" );
document.write( "B(1) = 6*(0.153^1)*(0.847)^5<BR>\n" );
document.write( "B(1) = 0.40018401105694<BR>\n" );
document.write( "B(1) = 0.400184\r<BR>\n" );
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document.write( "Add the results<BR>\n" );
document.write( "B(0)+B(1) = 0.369233 + 0.400184 = 0.769417<BR>\n" );
document.write( "The probability of at most 1 hit is roughly 0.769417\r<BR>\n" );
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document.write( "The complement of this is <BR>\n" );
document.write( "1-0.769417 = <font color=red>0.230583</font> which is the approximate final answer.<BR>\n" );
document.write( "The probability of at least 2 hits is roughly 0.230583  (i.e. about a 23.0583% chance).\r<BR>\n" );
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document.write( "Why did I subtract from 1?<BR>\n" );
document.write( "Well consider the fact that<BR>\n" );
document.write( "B(0) + B(1) + B(2) + B(3) + B(4) + B(5) + B(6) = 1<BR>\n" );
document.write( "All of the probability values of a distribution must add to 1.\r<BR>\n" );
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document.write( "We already calculated the stuff highlighted in blue<BR>\n" );
document.write( "<font color=blue>B(0) + B(1)</font> + <font color=red>B(2) + B(3) + B(4) + B(5) + B(6)</font> = 1<BR>\n" );
document.write( "and the stuff in red represents what we want to calculate.\r<BR>\n" );
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document.write( "We can subtract the blue stuff from both sides to isolate the red stuff.<BR>\n" );
document.write( "<font color=red>B(2) + B(3) + B(4) + B(5) + B(6)</font> = 1 - ( <font color=blue>B(0) + B(1)</font> )\r<BR>\n" );
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document.write( "---------------------\r<BR>\n" );
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document.write( "Here are a few binomial distribution calculators.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.gigacalculator.com/calculators/binomial-probability-calculator.php\">https://www.gigacalculator.com/calculators/binomial-probability-calculator.php</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/binomial-distribution\">https://www.omnicalculator.com/statistics/binomial-distribution</a><BR>\n" );
document.write( "You could also use a spreadsheet or a TI83/TI84 calculator.\r<BR>\n" );
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document.write( "Here is an article talking about binomial probabilities using a TI84<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">https://www.statology.org/binomial-probabilities-ti-84-calculator/</a><BR>\n" );
document.write( "For this particular problem, we can input <font color=red>1 - binomcdf(6, 0.153, 1)</font> into the TI84. <BR>\n" );
document.write( "This will add up the values from B(0) to B(1).<BR>\n" );
document.write( "Then subtract the result from 1 to get the final answer.<BR>\n" );
document.write( "Be sure to use the CDF and not the PDF.<BR>\n" );
document.write( "The PDF is one specific value, while the CDF adds up multiple values below a specific x value.\r<BR>\n" );
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document.write( "Another question involving the binomial probability distribution<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1201223.html\">https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1201223.html</a><BR>\n" );
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