x = ( ) 2 n C {H,T; H,H; T,H; T,T} ! r! 1 if outcome = H Toss a coin 0 if outcome = T 1 if there is a match 0 if there is no match = 36 outcomes

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1 Outcomes for tosses of a coin: () () ow many combinations of hamburgers ( thru F) and soft drins ( thru ) at Wendy's? - () () () here are x = possible combinations of hamburgers and soft drins () oss () oss () oss 8 Outcomes ow many different combinations of Wendy's hamburgers can we buy, if we buy burgers? (One combination could be the same burger twice) - No! From the top branch we can get combinations, but from the next branch only, the next only,, and = combinations, allowing the purchase of burgers of the same ind. n Number of Permutations of n things taen r at a time n! P r = = n n n n r+ ( n r)! different combinations? (x=?) What if we don't allow burgers of the same ind? Number of Combinations of n things taen r at a time n C r n n! npr = ( r ) = = r! ( n r)! r! hen, we can only find =! 7 = =!! combinations econd ie ample pace for One Roll of wo ie First ie = outcomes if outcome = x = if outcome = if there is a match x = if there is no match {,;,;,;,} x = x = oss a coin Matching Pennies

2 econd ie Elementary Events One Roll of wo ie First ie = elementary or simple events Value of RV Probability istribution for Roll of ice probability = { e, e, p, p,, e }, p } econd ie ample pace for One Roll of wo ie First ie ll Elementary Ā events where the sum of the dice = 7 he sample space,, contains all the elementary events he Event contains all elementary events corresponding to an inclusion rule. ll other elementary events are in the complement -bar.

3 = { x x or x } he intersection consists of all outcomes common to two events. he Union of two events & consists of all elementary events contained in either or. Pr ( ) = P + P( ) = Pr( or ) _ Pr{ } = Pr + Pr = Pr + Pr =. Ā _ and are two mutually exclusive events (no elementary events in common), Or, more generally: Pr( C ) = P + P( ) + P( C) + = Pr( or or Cor ) he Probability of the union of complementary events, & -bar is one! Pr( ) = P + P( ) P( ) = Pr( or ) Results of ntihistamine Experiment Received the rug Received the Placebo otal llergic ymptoms 8 No llergic ymptoms 7 otal 8 he General ddition Rule requires the subtraction of the probability of Intersecting events. or, more generally, Pr( C ) = P + P( ) + P( C) + P( C ) = Pr( or or Cor )

4 Contingency able Results of ntihistamine Experiment Received the Received the otal rug Placebo llergic ymptoms 8 No llergic 7 ymptoms otal 8 Cells Received the rug Received the Placebo P & P llergic ymptoms P ( ) =. P ( ) = 8. P=. No llergic ymptoms P =. P = P = 7. P Joint Probability istribution ( ) ( ). P& P = 8. P =.. otals llergic ymptoms No llergic ymptoms Received the rug Received the Placebo P ( ) =. ( ) P =. ( ) P = 8. ( ) P =. & P P P=. P = 7. P& P P = 8. P =.. Joint Probabilities Marginal Probabilities he Conditional Probability Rule: If and are any two events, then ( ) P P ( & ) =. P In words, for any two events the conditional probability that one event occurs given that the other even has occurred equals the joint probability of the two events divided by the probability of the given event. econd ie ample pace for One Roll of wo ie First ie P(C)= / = / ll Elementary events where the sum of the dice = 7. (Event C) econd ie ample pace for One Roll of wo ie First ie P(C ie =)=/ ll Elementary events where the first die = he only elementary event that leads to a 7 when the first die was a.

5 econd ie () Ways to Roll a 7 First ie () Ways to roll a when ie = he General Multiplication Rule: If and are any two events then P( & ) = P( ) P( ). In words, for any two events, their joint probability equals the probability that one of the events occurs times the conditional probability of the other event, given that event. P ( & ) P ( = = ). 8 P ( ) = = = = 7. = P ( = ) = PC ( = 7). P P ( = ) 7. Independent Events Event is said to be independent of event if the occurrence of event does not affect the probability that event occurs. In symbols, = P P. In words, nowing whether event has occurred provides no probabilistic information about the occurrence of event. Independence Implies P( ) P( ) = = P = P( ) P( ) P( ) P( ) = = P( ) = P( ) P Rearranging these two equations gives the General Multiplication Rule: P ( ) = PP ( ) or P ( ) = PP ( ) which implies, since P( ) = P( ) P( ) P( ) = P( ) P that, if and are independent events, then from the definition of independence: ( ) = = ( ) = ( ) = P PP P P or, P PP P P he pecial Multiplication Rule (for wo Independent Events) If and are independent events, then = P & P P, and conversely, if P & = P P, then and are independent events. In words, two events are independent if and only if their joint probability equals the product of their marginal probabilities. & & & ample space,, is completely filled by event.

6 & & & ample space,, is completely filled by event. ample space,, is completely filled by event. he Rule of otal Probability uppose that Events,,..., are mutually exclusive and exhaustive; that is, exactly one of the events must occur. hen, for any event, = ( j ) = ( j) ( j) P P P P. j= j= ayes's Rule: ssume these probabilities are nown(let = ): P P ( ) P ( ) ( ) ( ) ( ) P P P Use these probabilities to find the following probabilities: ( ) ( ) ( ) P P P From the Conditional Probability Rule, we now that: ( & i ) P( ) P P = for i =,. i Next, apply the General Multiplication Rule to the Numerator: hat gives us: Posterior probability ( & i ) P P( i ) = i=,. P Prior probability Finally, apply the rule of otal Probability in the denominator to give us ayes's Rule: ( i ) P( i) P( i) P( j) P( j) P = i =, j= ( i) P( i) P P( i ) = i =,. P

Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com

1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix