Probability three is 1/6. If necessary, you may need to simplify your fraction, but in this case the fraction is already in simplest form. Probability is a fraction where the Probability can be addressed, however, innumerator is the number of times the event a percent or a decimal along as well asoccurs and the denominator is the total number fractions. To determine the probability in aof trials. It is represented by number between 0 decimal, you must first determine what theand 1 (or 0% to 100%), and it is used to show fraction probability is, which is 1/6 in thisthe likelihood of an event’s occurrence. In example. Then, you divide the numerator byorder to find probability, you must first count the denominator, and your quotient should be ahow many trials there are and how many times decimal value, which is your decimal. If yourthe event occurs. If it doesn’t tell you how decimal is repeating or is very long, you maymany trials there are, you may need to add up want to consider rounding it to the nearestall of the events that occurred assuming it gives hundredth.you that data. The event that occurs is the thing It is possible to find a percent probabilitythat you want to happen, for example rolling a from both a decimal and a fraction. Tothree on a fair number cube. The amount of determine a percent from a decimal, all youtrials there are is the amount of outcomes that need to do is move the decimal 2 digits to thecan occur. When you roll a fair number cube, right. For example, with .200, you would movethe possible outcomes that can occur are the decimal place twice to the right, which gets1,2,3,4,5, and 6, a total of 6 possible you 20.0, 20.0%, or 20%. To determine aoutcomes. Since there are six possible probability from a fraction, you must set up aoutcomes (denominator) and one event that can proportion, one of your ratios being youroccur (numerator), the probability of rolling a fraction that you already know and the other
ratio being x over 100 (because a percent is a probability that she will pick a nine in afraction where the denominator is assumed as percent?being 100). Then, you must use the crossproduct method where you multiply the first Out of the 52 to cards, or the total number ofratio’s denominator by the second ratio’s trials, 4 events could occur, the four “9” cardsnumerator and the first ratio’s numerator by the in the deck (nine of hearts, diamonds, spades,second ratio’s denominator. One of your and clubs). The number of times the event canproducts out of these two values will be a occur is 4 (numerator), and the number of trialsconstant, but one will have a variable in it. or possible outcomes that could happen is 52These two values are equal, so you should be (denominator) because there are 52 differentable to make an equation that states c = cx cards Juanita could pick from the deck. We(where c stands for constant and x stands for now have the fraction 4/52. We must now setvariable). You can solve this equation by doing up a proportion indicating how many times thea one-step, where you will undo the all of the event would have occurred if there were 100values that restrict the variable to be isolated by trials to get a percent. Your two proportionaldoing the inverse operation. Then, to keep an ratios will be 4/52 and x/100. You will firstequal balance, you must perform the same cross-multiply the numerator of the first ratio byaction to the other side (the two sides are the denominator of the second ratio, and thoseseparated by the equal sign). You should now two numbers are 4 and 100, which equals 400.get answer that reads c = x (where c is a The other numbers you multiply are 52 and x,constant and x is a variable. Your constant which equals 52x. The value 52x is equal tovalue is your percent. the value 400, and we can represent thisProblem 1: Juanita randomly picks a card from through the equation 400 = 52x. We want toa standard 52-card deck. What is the
isolate x, so we will undo the operations that Experimental probability is determined byare restricting x from being isolated (that conducting an experiment repeatedly andoperation in this case is *52) by performing the observing the number of times the event occurs ininverse operation, which in this case is to divide the experiment compared to the total number of trials. For example, if you rolled a fair numberby 52. You now have isolated x, however you cube 5 times, and 3 out of the 5 times you rolled amust now divide 400 by 52 to keep both sides 2, your experimental probability of rolling a 2of the equation equal. You now have the would be 3/5. Experimental probability is differentanswer, 7.692307 repeating, which can be from theoretical probability because it is based offrounded to the nearest hundredth as 7.7, or of the outcome of an experiment that was done a7.7%. limited number of times. The theoretical probability on the other hand measures the probability of anProblem 2: John, Chase, Micah, Willie, and Ms. event occurring assuming the trial was performedHowson competed in a running race. All of the infinity times. In other words, theoretical5 runners have an equal chance of winning probability shows the likelihood of an eventexcept for Ms. Howson, who has twice as much occurring IF you performed an experiment, whileof a chance as winning. What is the probability experimental probabilities shows how often thethat Ms. Howson would not win the race? event occurred in an experiment THAT WAS ALREADY CONDUCTED compared to the total number of trials.4/6 or 2/3 Problem 1: D’Andrus flipped a penny 16 times, and 10 out of those 16 times she got “heads”. What is Experimental Probability the experimental probability that D’Andrus flipped a “tail”?
The experimental probability that D’Andrus flippeda head was 10/16, because in this experiment, he Theoretical Probabilityreceived the event heads 10 out of the 16 trials.The best way to find the number of times the event Theoretical probability is found bydid not occur is to subtract the numerator from the dividing the number of times an event coulddenominator using the fraction representing the potentially occur in an experiment by thenumber of times the event did occur. In this case, number of outcomes that could possibly happen16-10 is 6, and 6 will be our numerator. We will in the experiment based on what theuse the same denominator as we did for the experiment would look like if you repeated inprobability for the number of times D’Andrusflipped “heads” because the number of trials does infinity times. It is different from experimentalnot change. We now have the answer 6/16, which probability because it is based off of how manycan be simplified to 3/8. The experimental times an event COULD occur in the experiment,probability that D’Andrus flipped a “tail” is 3/8. not the number of times the event DID occur in the experiment. In other words, you areProblem 2: You spin a spinner with sectors labeled biasing theoretical probability based on odds,green, yellow, blue, orange, and red. You spin red not what happened in an experiment.once, yellow twice, blue three times, orange sixtimes, and red once. What was your experimental Problem 1: There are 9 blue sheets of paper,probability of spinning orange or blue? 10 green sheets of paper, 2 white sheets of paper, 4 yellow sheets of paper, and 5 black9/13 sheets of paper in a box and I pulled out one sheet of paper randomly, what is the theoretical probability that I would select a white or green sheet of paper?
Tree Diagrams and the Counting PrincipalTo find out the number of possible outcomes A tree diagram is a diagram that helpsthat could occur, you must add up all of the you see all of the possible outcomes in a set. Itevents that occurred and the number of times consists of branches and the categories increaseeach event could occur. When you add up all and get more specific as you go down. Anof the events and numbers of times each event example of a tree diagram is included below.could occur (9+10+2+4+5), you get a sum of30. Now you know that your probability’sfraction’s denominator is 30 because thenumber of trials in this set is 30. The favorableevents we want to occur for our numerator aregreen and white. White occurred twice, andgreen occurred 10 times. These two valuesmust be put together to reach 12 on thenumerator because 12 out of the 30 trials arefavorable events. We now have the fraction12/30, or 6/15, which is our probability.
The counting principal refers to how to find a*b because there are more events. Sincethe occurrence of more than one activity. each of the 10 events have 4 outcomes, weThe rule for the fundamental counting multiply 4*4*4*4*4*4*4*4*4*4, or 4 toprincipal is that if there are a possible ways the 10th power. The product is 1,048,576.to choose the first item and b possible ways This means that the odds of getting ato choose the second item (after the first perfect score on the test would beitem was chosen), then there are a*b ways 1/1,048,576. The numerator is 1 becauseto choose all of the items combined. This there is only one combination of answersalso works for more than one event. to choose that will get you a perfect score, or only one favorable outcome.Problem 1: If I am given a 10 questionmultiple choice (A, B, C, D) quiz, and I Problem 2: An ice cream shop offers 31guess on every single question, what is the flavors and 5 toppings. How manyprobability of scoring a perfect 100%? different types of ice cream could be made with one flavor and one topping?In this case, there are 10 events asopposed to 2, but the concept of the 155 different types of ice creamfundamental counting principal stillapplies. However, our expression will looklike: a*b*c*d*e*f*g*h*i*j instead of just
Independent and Dependent Events: Dependent Events There are 52 cards in a deck. What is the probability of getting two “9”s in a row? If the events are independent, the existence of If this was an independent event, I would multiply 4/52one event does not affect the probability of the other by 4/52, but this is a dependent event, meaning theevent. If the events are dependent, the existence of one outcome of the 1st event will affect the outcome of theevent does affect the probability of the other event. An second event. Since one card is being pulled out of theexample of an independent event is rolling a “3” on a deck, the second time I’m pulling out a card, there arefair number cube on one roll and a “4” on the next roll. only 51 cards or 51 trials, meaning my denominatorAn example of a dependent event is the amount of time changes to 51. There is also one less “9” cards in thea student spends studying for a test and the probability deck because I removed one of the “9” cards from theof them receiving a 90 or above on the test. deck when I picked my first card. Since there is one less event that can occur, my numerator changes to 3.Independent Events So, instead of doing 4/52*4/52, we will multiply 4/52*3/51. The product is 12/2652, which can beProblem 1: What is the probability of rolling a 6 on a simplified to 1/221.fair number cube twice in a row? Problem 2: A gumball machine has 20 red, 20 green,Since this is an independent event, you can multiply 20 blue, and 20 pink gumballs. Dennis then takes a1/6 by 1/6 (fundamental counting principal) to pink gumball from the machine. Then, Mark takes adetermine the answer. The answer is 1/12. gumball. What is the probability that the gumball he chooses will be red?Problem 2: What is the probability of tossing “heads”on a coin twice in a row? 20/791/4
Mark Winokur 9/23/11 Green Probability Guide BookProbability guide book By Mark Winokur