Probability intro.html (8272B)
1 <?xml version="1.0" encoding="UTF-8"?> 2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> 3 <html><head><link rel="stylesheet" href="sitewide.css"><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/><meta name="exporter-version" content="Evernote Mac 7.6 (457297)"/><meta name="altitude" content="-4.208069801330566"/><meta name="author" content="Alex Balgavy"/><meta name="created" content="2018-12-16 00:43:31 +0000"/><meta name="latitude" content="52.30035400390625"/><meta name="longitude" content="4.988170682800604"/><meta name="source" content="desktop.mac"/><meta name="updated" content="2018-12-16 01:27:31 +0000"/><title>Probability intro</title></head><body><h1>Probability intro</h1><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><div>sample space: set of all possible outcomes</div><div><span style="font-size: 16px;"> 4 <img src="Probability%20intro.resources/82DA0CAB-1ECF-4D59-9F2A-D1234AE1EA04.png" height="16" width="115"/></span></div></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><div>event: collection of outcomes (capital letters)</div><div><span style="font-size: 16px;"> 5 <img src="Probability%20intro.resources/E0C52A8D-5988-4E5F-9DD3-C583246D76B0.png" height="18" width="267"/></span></div></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><div>probability measure: value between 0 and 1</div><div><span style="font-size: 16px;"> 6 <img src="Probability%20intro.resources/BDA4CF06-F628-462F-8605-7C6BCD8F711E.png" height="34" width="153"/></span></div></div><ul><li><div>P(A) = 0: event is impossible</div></li><li><div>P(A) = 1: event is certain</div></li></ul><div><br/></div><h2>Determining probability</h2><ol><li><div>Estimate with relative frequency:</div></li></ol><div style="margin-top: 1em; margin-bottom: 1em; margin-left: 40px;-en-paragraph:true;"><span style="font-size: 16px;"> 7 <img src="Probability%20intro.resources/3FD28422-E813-4F7F-8E34-1F23E59D42C5.png" height="99" width="343"/></span><br/></div><ol start="2"><li><div>Theoretical approach: make a probability model</div></li><li><div>Subjective approach: estimate P(A) based on intuition/experience</div></li></ol><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Finding P(A) for discrete case:</div><ol><li><div>Find sample space <span style="font-size: 16px;"><img src="Probability%20intro.resources/35164666-FA75-49AB-A84D-6C6FD626E6D1.png" height="13" width="11"/></span></div></li><li><div>Determine probabilities 8 <img src="Probability%20intro.resources/DB38D8AB-300C-4A63-BDFC-96B0BCF653C0.png" height="16" width="30"/> for all 9 <img src="Probability%20intro.resources/C5248918-FC1B-4F58-A428-6AAE812CD000.png" height="13" width="37"/></div></li><ul><li><div>if all equally likely, then 10 <img src="Probability%20intro.resources/318A5966-6BCE-42F9-BE72-B9B36A6798D7.png" height="16" width="75"/> where N is number of outcomes in 11 <img src="Probability%20intro.resources/4FDC25D6-CF44-49AC-96BA-1C64C4C2795D.png" height="12" width="10"/></div></li></ul><li><div>Determine which outcomes are in A</div></li><li><div>Compute P(A) by </div></li></ol><div style="margin-left: 40px;"><span style="font-size: 16px;"><br/></span></div><div style="margin-left: 40px;"><span style="font-size: 16px;"> 12 <img src="Probability%20intro.resources/4F559F81-FF2E-497B-9DFA-EC210A10C01F.png" height="37" width="137"/></span></div><div style=""><font style="font-size: 14px;"><br/></font></div><h2>Probability rules:</h2><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">“At least one”:</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 13 <img src="Probability%20intro.resources/73C45B52-3233-4850-A675-FC9270B18B4C.png" height="18" width="210"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Addition rule (A and B):</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 14 <img src="Probability%20intro.resources/8947E446-03F4-4FE2-A919-B60AC846569F.png" height="18" width="268"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Complement (not A):</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 15 <img src="Probability%20intro.resources/994FA04E-5C98-4E29-8FED-B3F3C95B0563.png" height="19" width="121"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 16 <img src="Probability%20intro.resources/30C6C2F5-24D9-497C-8CF0-E5F30E56C453.png" height="19" width="212"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Conditional probability (B given A):</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 17 <img src="Probability%20intro.resources/10FF60DB-0B9C-42D1-8F63-1771C85CC905.png" height="40" width="145"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 18 <img src="Probability%20intro.resources/CD098AB5-85D4-400E-8AB7-8B06F5600D5E.png" height="18" width="199"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 19 <img src="Probability%20intro.resources/18BD58AF-261C-4798-95B0-1B0D1FD84316.png" height="19" width="144"/></span><br/></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 20 <img src="Probability%20intro.resources/B4FA55EC-6128-4BD7-B20F-607AD531BE96.png" height="19" width="156"/></span> <span style="font-weight: bold;"> (NOT IF COMPLEMENT IS IN CONDITION)</span></div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Disjoint events (mutually exclusive):</div><ul><li><div><span style="font-size: 16px;"> 21 <img src="Probability%20intro.resources/863448C3-05B6-42E8-99F5-EAD2A017F781.png" height="18" width="181"/></span></div></li><li><div><span style="font-size: 16px;"> 22 <img src="Probability%20intro.resources/E52D5FEF-A442-4256-83C9-190BC0FA2DE7.png" height="18" width="96"/></span></div></li></ul><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Independent events:</div><ul><li><div><span style="font-size: 16px;"> 23 <img src="Probability%20intro.resources/1FD39172-02F8-4B50-A391-53475092F1BB.png" height="18" width="181"/></span></div></li><li><div><span style="font-size: 16px;"> 24 <img src="Probability%20intro.resources/A1FE0B75-7AA4-49D8-9C1C-1830554F6ED3.png" height="18" width="111"/></span></div></li></ul><div><span style="font-size: 16px;"><br/></span></div><h2>Bayes theorem</h2><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">Forget that complicated-ass formula. You literally never need to use it.</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">For example, given these values:</div><ul><li><div><span style="font-size: 16px;"> 25 <img src="Probability%20intro.resources/224D8EB6-BD63-43E5-BDA1-E6EEF63D3988.png" height="18" width="86"/></span></div></li><li><div><span style="font-size: 16px;"> 26 <img src="Probability%20intro.resources/E67A439E-4A1D-4171-B02F-07C8B905048C.png" height="19" width="87"/></span></div></li><li><div><span style="font-size: 16px;"> 27 <img src="Probability%20intro.resources/9451F0BE-1227-40F8-90F9-38B0DA866992.png" height="18" width="97"/></span></div></li><li><div><span style="font-size: 16px;"> 28 <img src="Probability%20intro.resources/060DFC41-4A01-45EA-80A9-55FACD3529B3.png" height="19" width="105"/></span></div></li></ul><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;">You need to calculate 29 <img src="Probability%20intro.resources/846310A1-4C13-42A3-9F9E-33AD89AA7E0C.png" height="16" width="48"/>. Use conditional probability and do some rewriting:</div><div style="margin-top: 1em; margin-bottom: 1em;-en-paragraph:true;"><span style="font-size: 16px;"> 30 <img src="Probability%20intro.resources/C0823957-9584-4C60-A9E8-9D414EF6B99B.png" height="312" width="338"/></span><br/></div></body></html>