Probability, Events - Introduction to Statistics in Psychology - Lecture Slides, Slides of Statistics for Psychologists

Download Probability, Events - Introduction to Statistics in Psychology - Lecture Slides and more Slides Statistics for Psychologists in PDF only on Docsity! Introduction to Statistics in Psychology PSY 201 Lecture 14 probability Coincidences are rarely interesting. PROBABILITY number between 0 and 1 probability of event A is written as P (A) if P (A) = 1.0 it indicates with certainty that event A will happen if P (A) = 0 it indicates with certainty that event A will not happen 2 EVERYDAY EVENTS people often have misconceptions about the way probabilties interact things that seem rare may not actually be interesting to analyze the probability of events that seem unusual • Julius Ceasar • Hitting streaks • Predictive dreams • Shared birthdays • Con games with cards 3 JULIUS CAESAR Some 2000 years ago (or so) Julius Caesar is said to have gasped “You too, Brutus? Then I die.” as his friend stabbed him to death What are the chances that you just inhaled a molecule that came out of his mouth? Surprisingly good! Almost 0.99. Assumes • Caesar’s dying breath contained about A = 2.2 ! 1022 molecules • Those molecules are free and distributed around the globe evenly. • Your inward breath contained about B = 2.2! 1022 molecules • The atmosphere contains about N = 1044 molecules 4 JULIUS CAESAR If there are N molecules and Caesar exhaled A of them, then the probability that any given molecule you inhale is from Caesar is P (m from C) = A N = 2.2"23 which is very small! So the probability that any given molecule you inhale is not from Caesar is the complement: P (m not from C) = 1"A N = 1"2.2"23 5 JULIUS CAESAR So the probability of inhaling B molecules that are not from Caesar is P (breath not from C) = ! """#1" A N $ %%%& B # 0.01 So the probability of your breath containing a molecule from Caesar is approximately 1-0.01 = 0.99! 6 docsity.com HITTING STREAKS Pete Rose set a National League record with 44 consecutive games with a safe hit this is impressive, but is it rare? Rose batted around 0.300 (had a safe hit 30% of the time) so, assuming 4 at bats per game, the probability of not getting a hit during a game is P (no hit) = (1" 0.3)4 = 0.24 So the probability of getting at least one hit is 1-0.24 = 0.76. 7 HITTING STREAKS Still, the probability of getting hits in any given sequence of 44 games is P (44 streak) = (0.76)44 = 0.000005699 and the probability of not getting a streak is P (not 44 streak) = 1"(0.76)44 = 0.999994301 8 HITTING STREAKS But there are 162 games in a season, so there are 118 sets of 44 consecutive games. Thus, the probability of not getting a streak of hits in at least 44 consecutive games out of a 162 game season is: P (no streak) = (0.999994)118 = 0.999327 so the probability of a 44-game streak is P (streak) = 1"(0.999994)118 = 0.000672 (includes the possibility of streaks of more than 44 games) Still very rare! 9 HITTING STREAKS But how many players have been in the Major Leagues at any given time? (say 30 that bat like Rose) the probability that every player will not get a streak of at least 44 games in a given year is P (no streak) = (0.9993)30 = 0.9800 So probability that at least one player gets such a streak is 1.0" 0.980027651 = 0.019972349 still small! 10 HITTING STREAKS And how many years has baseball been played? (say 100) the probability that every year everyone will not get a streak of at least 44 games in a given year is P (no streak) = (0.9800)100 = 0.1329 So probability that at least one player on some year gets such a streak is 1.0" 0.132994269 = 0.867005731 which is pretty good odds! Thus, we can expect that Rose’s streak will be broken eventually (unless pitchers become much better) 11 PREDICTIVE DREAMS ever dream something and had it come true? Many people take this occurence as evidence of extrasensory perception and “other worlds”. But it’s actually not that uncommon from a probabilitistic point of view suppose that the probability that a night’s dream matches some later event in life is 1 in 10000 P (predictive dream) = 0.0001 Then the chance that a dream is non-predictive is P (non predictive dream) = 1"0.0001 = 0.9999 assume that dream predictiveness is independent 12 docsity.com