Comparing & Contrasting. - mathematically. Ways of comparing mathematical and scientific quantities...

Transcription

1 Comparing & Contrasting - mathematically Ways of comparing mathematical and scientific quantities...

2 Comparison by Division: using RATIOS, PERCENTAGES, ODDS, and PROPORTIONS

3 Definition of Probability: It is a numerical measure of the likelihood that a specific event will occur. The probability of an event happening is measured by a number between 0 and 1. Notation and Interpretation If E is an event, then the probability of E happening is denoted by: P(E). P(E) = 0 means that E will definitely NOT happen. It is an impossible event. P(E) = 1 P(E) = 0.5 means that E will definitely happen. It is a sure event. means that there is a chance of E happening.

4 Probability of an Event happening P(E) = 0 means that E will definitely NOT happen. It is an impossible event. The closer that P(E) is to 0, the less likely it is to occur. o The probability that summer temperatures will reach 42 Celsius at the geographic North Pole is pretty close to 0 - at least for now. P(E) = 1 means that E will definitely happen. It is a sure event. The closer that P(E) is to 1, the more likely it is to occur. The probability of rain in India during the monsoon season is probably 1.

5 Probability of an Event NOT happening If E is an event, then the probability of E happening is denoted by: P(E) and the probability of E NOT happening: 1 - P(E). If the weather forecast predicts a 40% chance of rain tomorrow, then the probability that it will rain tomorrow (event E) is 40% written as a decimal. Therefore, P(E) = 0.40 and the probability that it will NOT rain tomorrow is: P(not E) = 1 - P(E) = = Note that P(E) + P(not E) = P(E) P(E) = 1 because it is absolutely certain that either it will rain or it won t.

6 The Odds for an Event Happening If E is an event having probability P(E), then the odds for E happening is defined to be the ratio P(E) : ( 1 - P(E) ) or P(E) : P( not _ E ) or P(E) : P( E ). The Odds against an Event Happening If E is an event having probability P(E), then the odds against E happening is defined to be the ratio ( 1 - P(E) ) : P(E) or P(not E) : P(E) _ or P( E ) : P(E). _ The event that E will not happen is generally denoted by E.

7 If the weather forecast predicts a 40% chance of rain tomorrow, then the probability that it will rain tomorrow (event E) is P(E) = 0.4 and the probability that it will _ not rain tomorrow (event E ) is P( E ) = 1 - P(E) = 0.6. _ Therefore, the odds against it raining tomorrow are: ( 1 - P(E) ) : P(E) or.6 :.4. The odds against ratio expressed as a ratio fraction is: _. ( 1 - P(E) ) or P(E).6.4 = 3 2 Interpreted this means that there are 3 chances to 2 that it will not rain tomorrow or, reading the ratio in reverse, there are 2 chances to 3 that it will rain tomorrow.

8 The odds for it raining tomorrow are expressed by the reverse ratio of the odds against ratio. P(E) : ( 1 - P(E) ) or.4 :.6. The odds for ratio expressed as a ratio fraction is: P(E) or.4 ( 1 - P(E) ).6 _ = 2 3 Interpreted this means that there are 2 chances to 3 that it will rain tomorrow, simply the reverse reading of the odds against ratio.

9 Statistical Odds versus Betting Odds Before the 2009 Stanley Cup playoff season had begun, the oddsmakers had listed the Detroit Red Wings as 5-1 favourites and the Pittsburgh Penguins as 6-1 favourites to win the Cup. The Ducks were 8-1 favourites. The Bruins, Hurricanes, Blackhawks, Canucks, and Capitals were all listed as having odds to win somewhere between 15-1 and These are betting odds. The question is: Do these odds reflect real statistical probabilities?

10 Random Choice Probabilities At the beginning of the second round of the playoffs, the eight remaining teams in contention for the Cup were: the ANAHEIM DUCKS, the BOSTON BRUINS, the CAROLINA HURRICANES, the CHICAGO BLACKHAWKS, the DETROIT RED WINGS, the PITTSBURGH PENGUINS, the VANCOUVER CANUCKS, and the WASHINGTON CAPITALS. Without knowing anything about hockey or the teams performances throughout the season, one could say: Well, there are 8 teams and only one team can win. Therefore, each team has a probability of 1/8 of winning the Cup. This choice is tantamount to a random picking of the name of one of the teams out of a hat containing the 8 names.

11 Random Choice Odds By random choice, we have just determined that each of the second round playoff teams had a probability of 1/8 to win the Stanley Cup. How does a probability of 1/8 convert into the random choice statistical odds of any one of the teams winning? If E is an event having probability P(E), happening is defined to be the ratio _ P(E) = 1/8 P( E ) = _ P(E) E 1 - P(E) = then the odds for E : P( ). 1-1/8 = The odds for any one team _ winning were: P(E) : P( E ) 1/8 : 7/8 1 : 7 Therefore, the random choice statistical odds for any one of the 8 teams to win the 2009 Stanley Cup was: 1 : 7 7/8

12 Statistical Ratios: Goals Scored / Games Played Ratios We all know that some teams are better than others, so the odds of winning created by random choice are not intelligent odds. Let s now take a look at the teams goals scored to games played ratios giving average goals scored per game for the 2009 hockey season - in descending order. DETROIT RED WINGS 289 / 82 = 3.52 BOSTON BRUINS 270 / 82 = 3.29 WASHINGTON CAPITALS 268 / 82 = 3.27 CHICAGO BLACKHAWKS 260 / 82 = 3.17 PITTSBURGH PENGUINS 258 / 82 = 3.15 VANCOUVER CANUCKS 243 / 82 = 2.96 ANAHEIM DUCKS 238 / 82 = 2.90 CAROLINA HURRICANES 236 / 82 = 2.88 Some oddsmakers might include these and other such ratios in their calculations about what betting odds to offer, but statistical. probabilities they are not. Note that the sum of the numerator and denominator parts of the ratio does not equal statistical certainty, which in unreduced statistical odds is 1.

13 a ratio of probabilities - except in the betting world! However, some would venture to say that probabilities of winning in any sport are only of real importance if there is money involved. Enter the oddsmakers. Betting Odds: Futures Bets These are bets about something that will happen in the future - usually the winning of a Championship. The odds for the betting are set by oddsmakers. What is certain is that the more likely the chance of an event happening (that is, the closer the statistical probability of the event is to 1), the worse the payout is for the bettor. Conversely, the less likely chance of an event happening (or the closer the statistical probability of the event is to 0) the bigger the payout.

14 a ratio of probabilities - except in the betting world! Betting Odds: Futures Bets In July 2009, the odds on the New York Yankees winning the 2009 World Series were 4/1 and those of the Washington Nationals were 5000/1. While these fractional betting odds may look like statistical odds, they are not ratios of probabilities. They are ratios signifying payouts in the case of a win. Suppose you were to place a $100 bet on both teams. What would the payout be if the Washington Nationals were to actually win the World Series in 2009? Betting Odds Bet Amount Payout 5000 / 1 $ / 1 $100 $5000 $500,000 Not a bad payout for a long shot!

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