Chapter 6 Probability The study of randomness Warmup
Chapter 6: Probability The study of randomness
Warm-up 6. 4 MATCHING PROBABILITIES Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event. (The probability is usually a much more exact measure of likelihood than is the verbal statement. ) 0, 0. 01, 0. 3, 0. 6, 0. 99, 1 (a) This event is impossible. It can never occur. 0 (b) This event is certain. It will occur on every trial of the random phenomenon. 1 (c) This event is very unlikely, but it will occur once in a while in a long sequence of trials. 0. 01 (d) This event will occur more often than not. 0. 6
6. 8 THREE OF A KIND You read in a book on poker that the probability of being dealt three of a kind in a five-card poker hand is 1/50. Explain in simple language what this means. In the long run, of a large number of hands of five cards, about 2% (one out of 50) will contain a three of a kind. (Note: This probability is actually 88 /4165= 0. 02113. )
6. 1 The Idea of Probability Proportion of heads to tails in a few tosses will be erratic but after thousands of tosses will approach the expected. 5 probability
Random - does not mean haphazard - there are uncertain outcomes in the short run, but there is a pattern in the long run Ex: Random Digit Table From one digit to the next there is no way to tell what is going to be next, but in the table as a whole you would expect approximately 10% “ 0”, 10% “ 1”, 10% “ 2”, etc…
Probability - proportion of desired outcome in a large number of repetitions - long run relative frequency - a number between 0 and 1
Short-run Regularity o Ex: Flipping a coin 6 times o Most people would think that H T T H is more likely than T T T H H H but both are equally likely. o Ex: Getting T T T does not make it any more likely that you will get H on the next flip. o - Coins have no memory. - Each flip is an independent event. o
Streaks o If a basketball player makes several consecutive shots, people will say he is more likely to make his next shot because he is on a streak. o This is not true because making 8 shots in a row is no more likely than making 7 and missing 1. o Probability doesn’t show up in the short-run (one game). The statistics come from a whole lifetime of playing. o Players perform consistently, not in streaks.
Surprise Meeting o Running into someone in Times Square that you haven’t seen in 10 years might seem like a big coincidence. o But the average person has over 1500 acquaintances. The chance of running into a particular one of them is small, but running into any one of them is not that small.
Law of Averages o If a couple already has 4 girls, should there next child be a boy? o No, there is still a 50% chance of either. o In the long run (the population on the planet) we expect about half girls, half boys being born. But one family is a short run.
Long Run o If you flip a coin and get T T T, you won’t necessarily get H next. The law of averages only works in the long run. o o In the long run, if you flip a coin 10, 000 times, you will get about 50% H/T, but those first 5 T are overwhelmed by the next 9, 995 flips – the are not “compensated” for.
6. 2 Probability Models Probability A models have two parts: list of possible outcomes A probability for each outcome.
Sample Space To specify S we must state what constitutes an individual outcome, then which outcomes can occur (can be simple or complex) Ex: coin tossing, S = {H, T} Ex: US Census: If we draw a random sample of 50, 000 US households, as the survey does, the S contains all 50, 000
Rolling two dice At a casino- 36 possible outcomes when we roll 2 dice and record the up-faces in order (first die, second die) Gamblers care only about number of dots face up so the sample space for that is: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Techniques for outcomes 1. Tree diagram For tossing a coin then rolling a die finding
2. Multiplication Principle 2 x 6 3. = 12 for same example Organized list: H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5, T 6
With/without replacement If you take a card from a deck of 52, don’t put it back, then draw your 2 nd card etc. , that’s without replacement. Ex: how many different 3 digit numbers can you make: 10 x 9 x 8 = 720 If you take a card, write it down, put it back, draw 2 nd card etc. , that’s with replacement. Ex: 10 x 10 = 1000
Probability Rules 1. Any probability is a number between 0 and 1 2. The sum of the probabilities of all possible outcomes = 1 3. If 2 events have no outcomes in common (they can’t occur together), the probability that one OR the other occurs is the sum of their individual probabilities Ex: If one event occurs in 40% of all trials, another event happens in 25% of all trials, the 2 can never occur together, then one or the other occurs on 65% of all trials 4. The probability that an event doesn’t occur is 1 minus the probability that it does occur Ex: If an event happens in 70% of all trials, it fails to occur in the other 30%
Venn diagrams help! Ex: Probability of rolling a 5? B/c P(roll a 5 with 2 die) = P(1, 4) + P(3, 2) + P(2, 3) + P(4, 1) = 1/36 + 1/36 = 1/9 or. 111
Independence & Multiplication Rule the To find the probability for BOTH events A and B occurring Example: Suppose you plan to toss a coin twice, and want to find the probability of rolling a head on both tosses. A = first toss is a head, B = second toss is a head. So (1/2) = ¼. We expect to flip 2 heads on 25% of all trials. The more times we repeat this, the closer our average probability will get to 25%. The multiplication rule applies only to independent events; can’t use it if events are not independent!
Independent or not? Coin toss I: Coin has no memory and coin tossers cannot influence fall of coin Drawing NI: First pick, probability of red is 26/52 or. 5. Once we see the first card is red, the probability of a red card in the 2 nd pick is now 25/51 =. 49 Taking from deck of cards NI an IQ test twice in succession
More applications of Probability Rules If two events A and B are independent, then their complements are also independent. Ex: 75% of voters in a district are Republicans. If an interviewer chooses 2 voters at random, the probability that the first is a Republican and the 2 nd is not a republican is. 75 x. 25 =. 1875
Homework 11, 14, 15, 17, 18, 19, 20, 23 – 26, 27, 29 -31, 33, 35, 37
6. 3 General Probability Rules
Addition Rule for Disjoint events
General Addition rule for Unions of 2 events
Example: Deb and Matt are waiting anxiously to hear if they’ve been promoted. Deb guesses her probability of getting promoted is. 7 and Matt’s is. 5, and both of them being promoted is. 3. The probability that at least one is promoted =. 7 +. 5. 3 which is. 9. The probability neither is promoted is. 1. The simultaneous occurrence of 2 events (called a joint event, such as deb and matt getting promoted) is called a joint probability.
Conditional Probability The probability that we assign to an event can change if we know some other event has occurred. P(A|B): Probability that event A will happen under the condition that event B has occurred. Ex: Probability of drawing an ace is 4/52 or 1/13. If your are dealt 4 cards and one of them is an ace, probability of getting an ace on the 5 th card dealt is 3/48 or 1/16 (conditional probability- getting an Ace given that one was dealt in the first 4).
In words, this says that for both of 2 events to occur, first one must occur, and then, given that the first event has occurred, the second must occur.
Remember: B is the event whose probability we are computing and A represents the info we are given.
Extended Multiplication rules The union of a collection of events is the event that ANY of them occur The Intersection of any collection of events is the event that ALL of them occur
Example Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these only 1. 7% enter major league professional sports. About 40% of the athletes who compete in college and then reach the pros have a career of more than 3 years. Define these events: A = competes in college P(A) =. 05 P(B|A) =. 017 P(C|A and B) =. 400 What is the probability a HS athlete will have a pro career more than 3 years? The probability we want is therefore B = competes pro C = pro career longer than 3 years P(A and B and C) = P(A)P(B|A)P(C|A and B) =. 05 x. 017 x. 40 =. 00034 So, only 3 of every 10, 000 high school athletes can expect to compete in college and have a pro career of more than 3 years.
Extended tree diagram + chat room example 47% of 18 to 29 age chat online, 21% of 30 to 49 and 7% of 50+ Also, need to know that 29% of all internet users are 18 -29 (event A 1), 47% are 30 to 49 (A 2) and the remaining 24% are 50 and over (A 3). What is the probability that a randomly chosen user of the internet participates in chat rooms (event C)? Tree diagram- probability written on each segment is the conditional probability of an internet user following that segment, given that he or she has reached the node from which it branches. (final outcome is adding all the chatting probabilities which =. 2518)
Bayes Rule Another question we might ask- what percent of adult chat room participants are age 18 to 29? P(A 1|C) = P(A 1 and C) / P(C) =. 1363/. 2518 =. 5413 *since 29% of internet users are 18 -29, knowing that someone chats increases the probability that they are young! Formula sans tree diagram: P(C) = P(A 1)P(C|A 1) + P(A 2)P(C|A 2) + P(A 3)P(C|A 3)
6. 3 Need to Know summary(print) Complement of an event A contains all outcomes not in A Union (A U B) of events A and B = all outcomes in A, in B, or in both A and B Intersection(A^B) contains all outcomes that are in both A and B, but not in A alone or B alone. General Addition Rule: P(AUB) = P(A) + P(B) – P(A^B) Multiplication Rule: P(A^B) = P(A)P(B|A) Conditional Probability P(B|A) of an event B, given that event A has occurred: P(B|A) = P(A^B)/P(A) when P(A) > 0 If A and B are disjoint (mutually exclusive) then P(A^B) = 0 and P(AUB) = P(A) + P(B) A and B are independent when P(B|A) = P(B) Venn diagram or tree diagrams useful for organization.
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