Probability: introduction


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1 May 6, 2009 Probability: introduction page 1 Probability: introduction Probability is the part of mathematics that deals with the chance or the likelihood that things will happen The probability of an event is a measure of how likely it is that something will happen, expressed on a 0 to 1 scale where 0 means impossible and 1 means certain Basic probability calculations Many random situations involve having several outcomes that are equally likely For example, when you flip a coin, heads and tails are equally likely outcomes Here are two basic methods for computing probabilities, in situations where the outcomes are equally likely Method: If there are N equally likely outcomes, the probability of each of them is N1 Example: If you roll a 6sided die, the probability of rolling any specific number is 61 Method: An event is a set of outcomes If all outcomes are equally likely, the probability of an event is found using this formula: number of outcomes in the event Probability total number of outcomes Example: If you roll a 6sided die, what is the probability of rolling an even number? There are 3 even number outcomes (2, 4, 6) out of 6 total outcomes, so Probability 63 Often a probability fraction can be reduced, reexpressed as a decimal, or reexpressed as a percent In the example, 6 3 could be reduced to 21, written as a decimal 05, or written as 50% Unless you re given specific directions, any of these answer forms are acceptable Problems 1 You roll a 6sided die (labeled 1, 2, 3, 4, 5, 6) What is the probability of each of these? a probability of rolling a 1 b probability of rolling a 5 c probability of rolling an odd number d probability of rolling a 3 or higher e probability of rolling anything except 6 2 You pick a capital letter from the alphabet at random What is the probability of each of these? a probability of picking a Q b probability of picking the first letter of your own last name c probability of picking a vowel d probability of picking a consonant e probability that the letter can be written with straight lines only (no curvy lines)
2 May 6, 2009 Probability: introduction page 2 3 You have seven tiles from the game Scrabble that look like this: A L G E B R A Suppose you pick one of these tiles at random Find these probabilities a probability of picking an A b probability of picking a B c probability of picking a C d probability of picking anything except a G e probability of picking a vowel 4 A deck of cards has 52 cards These cards have different suits and colors as follows: 13 hearts ( ) which are red 13 diamonds ( ) which are red 13 spades ( ) which are black 13 clubs ( ) which are black Suppose you pick a single card at random Find these probabilities a probability of picking a heart ( ) b probability of picking a black card c probability of picking anything except a diamond ( ) d probability of picking the single card called the Ace of Spades (A ) 5 Here are the numbers of students in each grade at Jefferson High School Grade 9 (freshman) 23 Grade 10 (sophomore) 215 Grade 11 (junior) 220 Grade 12 (senior) 228 Suppose you pick a student from this school at random Find these probabilities a probability of picking a freshman b probability of picking a junior or a senior c probability of picking someone who is not a senior
3 May 6, 2009 Probability: introduction page 3 Probabilities involving or Often you will be asked to find the probability of one event happening or another event happening Problem 5b was an example of this: finding the probability of picking a junior or a senior There were actually two ways to do this problem Which of these two ways did you do it? Method 1: Add the number of juniors and the number of seniors ( ) 448 Then, divide by the total number of students: Method 2: Separately calculate the probability of picking a junior ( 900 ) and the probability of picking a senior ( 900 ) Then add fractions to get the answer: Here s a more general description of these methods and the situation where you can use them Suppose A and B are two events that are mutually exclusive (that is, it is impossible for both events to happen, only one or the other can happen) Here are two ways to find the probability that either A or B happens: Method 1: Add the number of outcomes in event A to the number of outcomes in event B Then, divide by the total number of possible outcomes Method 2: probability of A or B (probability of A) (probability of B) Probabilities involving not Often you will be asked to find the probability of an event not happening Problem 5c was an example of this: finding the probability of picking someone who is not a senior Here are two of the ways you could have done that problem: Method 1: Subtract the number of seniors from the total number of students ( ) Then, divide by the total number of students: Method 2: Calculate the probability of picking a senior ( 900 ) To get the probability of not picking a senior, subtract from 1: Here s a more general description of these methods and the situation where you can use them Suppose A is an event Here are two ways to find the probability that A does not happen: Method 1: Subtract the number of outcomes in event A from the total number of possible outcomes Then, divide by the total number of possible outcomes Method 2: probability of not A 1 (probability of A) Why learn two different methods for doing these or and not problems? In some problems, the information you have might only allow you to use one of the methods You ll see examples of this in the problems that follow
4 May 6, 2009 Probability: introduction page 4 Problems 6 Here are three of the spring girls sports at this high school: softball, lacrosse, tennis It is only possible to play one of these sports Suppose that there are 1000 girls in the school, and suppose that 60 of them play softball, 30 of them play lacrosse, and 50 of them play tennis Now suppose that one girl in the school is chosen at random Write answers to the following probability questions: first as fractions, then rewrite as decimals a Find the probability that the girl plays lacrosse b Find the probability that the girl does not play lacrosse c Find the probability that the girl plays lacrosse or tennis d Find the probability that the girl plays softball or tennis The manufacturers of M&M candies have released this information about the color proportions in a bag of candy Each number represents the probability that a randomly chosen M&M would have a particular color brown red yellow green orange tan Suppose that you select a single M&M from a newlyopened bag Find these probabilities: a probability that it is not brown b probability that it is yellow or green c probability that it is orange or red d probability that it is neither orange nor red Hint: use your answer from part c
5 May 6, 2009 Probability: introduction page 5 8 Again consider picking a card from the deck of cards described in problem 4 Write your answers to these probability questions as fractions a What is the probability of picking a heart or a diamond? b What is the probability of picking a club or a red card? c What is the probability of not picking a spade? 9 Suppose that for each car reaching a certain intersection, there is probability 055 that the car will go straight, probability 015 that the car will turn left, and probability 025 that the car will turn right a What is the probability that the car will not go straight? b What is the probability that the car will turn left or turn right? c Parts a and b should have the same answer Why? 10 There are 4 kinds of trains on Boston s MBTA Green Line, labeled B, C, D, and E Suppose that 30% of the trains are B trains, 2% are C s, 23% are D s, and 20% are E s Now suppose that you enter Park Street station and look at the first Green Line train that arrives Write your answers to these probability questions as decimals a Only the E trains go to Heath Street What is the probability that the train goes to Heath Street? b Only the D trains go to Fenway Station What is the probability that the train goes to Fenway Station? c To go to Brighton, you can take either a B train or a C train What is the probability that the train goes to Brighton? d To go to Kenmore Square, you can take any train that is not an E train What is the probability that the train goes to Kenmore Square?
6 May 6, 2009 Probability: introduction page 6 11 Whenever Taylor takes a math test, there is a probability of 4 1 that he gets an A and a probability of 04 that he gets a B Suppose that Taylor just took a math test Find each of these probabilities about his grade on that test a probability that Taylor did not get an A on the test b probability that Taylor did not get an B on the test c probability that Taylor did get an A or a B on the test d probability that Taylor s test grade was neither A nor B There was an important detail about the method of finding an or probability by adding: the events have to be mutually exclusive (meaning that there s no overlap between them) If the events aren t mutually exclusive, then adding won t give the correct answer 12 Again consider picking a Scrabble tile at random from this set: A L G E B R A Decide whether each of these probability calculations is correct or incorrect If incorrect, what is the correct answer? 2 3 a The probability of drawing an A or a B is b The probability of drawing an A or a consonant is c The probability of drawing an A or a vowel is d The probability of drawing an B or a consonant is e The probability of drawing a vowel or a consonant is 1
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