49207908-slides - Lecture Notes on PROBABILITY and...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture Notes on PROBABILITY and STATISTICS Eusebius Doedel http://cmvl.cs.concordia.ca/courses/comp-233/winter-2011/
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
SAMPLE SPACES DEFINITION : The sample space is the set of all possible outcomes of an experiment. EXAMPLE: When we flip a coin then sample space is S = { H , T } , where H denotes that the coin lands ”Heads up” and T denotes that the coin lands ”Tails up”. For a ” fair coin ” we expect H and T to have the same ” chance ” of occurring, i.e. , if we flip the coin many times then about 50 % of the outcomes will be H . We say that the probability of H to occur is 0.5 (or 50 %) . The probability of T to occur is then also 0.5. 1
Image of page 2
EXAMPLE: When we toss a fair die then the sample space is S = { 1 , 2 , 3 , 4 , 5 , 6 } . The probability the die lands with k up is 1 6 , ( k = 1 , 2 , · · · , 6). When we toss it 1200 times we expect a 5 up about 200 times. The probability the die lands with an even number up is 1 6 + 1 6 + 1 6 = 1 2 . 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
EXAMPLE: When we toss a coin 3 times and record the results in the sequence they occur, then the sample space is S = { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } . Elements of S are ” vectors ”, ” sequences ”, or ” ordered outcomes ”. We may expect each of the 8 outcomes to be equally likely. Thus the probability of the sequence HTT is 1 8 . The probability of a sequence to contain precisely two Heads is 1 8 + 1 8 + 1 8 = 3 8 . 3
Image of page 4
EXAMPLE: When we toss a coin 3 times and record the results without paying attention to the order in which they occur, e.g. , if we only record the number of Heads, then the sample space is S = { H, H, H } , { H, H, T } , { H, T, T } , { T, T, T } . The outcomes in S are now sets ; i.e. , order is not important. Recall that the ordered outcomes are { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } . Note that { H, H, H } corresponds to one of the ordered outcomes, { H, H, T } ,, three ,, { H, T, T } ,, three ,, { T, T, T } ,, one ,, Thus { H, H, H } and { T, T, T } each occur with probability 1 8 , while { H, H, T } and { H, T, T } each occur with probability 3 8 . 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Events In Probability Theory subsets of the sample space are called events . EXAMPLE: The set of basic outcomes of tossing a die once is S = { 1 , 2 , 3 , 4 , 5 , 6 } , so the subset E = { 2 , 4 , 6 } is an example of an event. If a die is tossed once and it lands with a 2 or a 4 or a 6 up then we say that the event E has occurred . We have already seen that the probability that E occurs is P ( E ) = 1 6 + 1 6 + 1 6 = 1 2 . 5
Image of page 6
The Algebra of Events Since events are sets , namely, subsets of the sample space S , we can do the usual set operations : If E and F are events then we can form E c the complement of E E F the union of E and F EF the intersection of E and F We write E F if E is a subset of F . REMARK. In Probability Theory we use E c instead of ¯ E , EF instead of E F . 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
If the sample space S is finite then we typically allow any subset of S to be an event. EXAMPLE: If we randomly draw one character from a box contain- ing the characters a , b , and c , then the sample space is S = { a , b , c } , and there are 8 possible events, namely, those in the set of events E = { } , { a } , { b } , { c } , { a, b } , { a, c } , { b, c } , { a, b, c } .
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern