Presentation on theme: "P ROBABILITY Mathematics is always thought of as being a study of certainties – the answer is either right or wrong. But there is a type of mathematics."— Presentation transcript:
P ROBABILITY Mathematics is always thought of as being a study of certainties – the answer is either right or wrong. But there is a type of mathematics which deals with uncertainties. Since much of our everyday life deals with uncertainties, mathematics has a method for describing these events. This branch of mathematics is called probability theory.
P ROBABILITY VOCABULARY Compound events Dependent events Event Experimental probability Independent events Outcome Probability Random Sample space Theoretical probability Visit the website listed below for definitions of these words: www.mathwords.com
P ROBABILITY SCALE The probability that an event will happen is somewhere between 0 and 1. When an event has no chance of occurring, we say its probability is zero. When an event is certain to occur, we say its probability is one (or 100%). Probabilities are usually written as a fraction or whole number is simplest form.
H OW TO FIND SIMPLE PROBABILITY To find the probability of an event when all outcomes are equally likely, use the formula: P = number of favorable outcomes number of possible events
E XAMPLE A standard deck of cards contains 52 cards. There are four groups of 13 cards. Each group is called a suit. Two of these suits are red and two are black. Each suit contains an ace, king, queen, jack, and the numbers 10 through 2. If you draw a card at random from the deck of cards, the probability of drawing a heart is… Because there are 13 hearts & 52 total cards.
C OMPOUND EVENTS A compound event consists of two or more simple events. To find the probability of compound events, you find the probability of each event and multiply them. The probability of drawing an Ace from a deck of cards AND rolling an odd number on a standard die is…
INDEPENDENT EVENTS When one event does not affect the others, we say that these are independent events. The probability of getting four tails in a row when a coin is tossed four times is 1 out of 16. These events are independent since one toss of the coin does not affect the outcome of the next toss. There are two possible outcomes, heads or tails, so the probability of getting a tail on each toss is ½. The probability of getting four tails:
P RACTICE PROBLEM #1 At Reyna High School 50% of the students eat lunch in the cafeteria. In the same school 10% of the students participate in sports. What is the probability that a student selected at random eats in the school cafeteria and participates in sports? A.C. B.D.
D EPENDENT EVENTS If the outcome of an event does affect the outcome of another event, we say that these are dependent events. Rick takes two marbles from a bag containing 3 red, 4 blue, 5 green, & 2 yellow marbles. What is the probability that both marbles are red? P(2 red) = P(red on 1 st draw) × P(red on 2 nd draw)
P RACTICE P ROBLEM #2 A jar contains 6 red marbles and 10 blue marbles, all of equal size. If Dominic were to randomly select 1 marble without replacement and then select another marble from the jar, what would be the probability of selecting 2 red marbles from the jar? A.C. B.D.
A DDITIONAL RESOURCES Additional resources for finding probability of compound events: http://www.mathgoodies.com/lessons/vol6/indepe ndent_events.html http://www.mathgoodies.com/lessons/vol6/indepe ndent_events.html
T HEORETICAL P ROBABILITY Theoretical probability is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is: P(event) = number of favorable outcomes number of possible outcomes. Examples of Theoretical Probability From the letters A, E, I, O, U the theoretical probability of selecting the letter E is.
P RACTICE PROBLEM #3 A coin is tossed on a standard 8×8 chessboard. What is the theoretical probability that the coin lands on a black square? A. 0.5 B. 0.25 C. 0.42 D. 0.6
E XPERIMENTAL P ROBABILITY Experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials. Examples of Experimental Probability Sam rolled a number cube 50 times. A 3 appeared 10 times. Then the experimental probability of rolling a 3 is 10 out of 50 or 20%. http://www.northstarmath.com/sitemap/experimentalprobability.html
P RACTICE P ROBLEM #4 A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads. A. C. B. D.
P RACTICE PROBLEM #5 The table shows the results of a number cube being rolled. Based on these results, what is the experimental probability of rolling a 1? A. 2.5% B. 1/6 C. 2/5 D. 0.6 OutcomeFrequency 16 22 32 43 52 60
P RACTICE PROBLEM #6 The table shows the results of rolling a fair number cube 50 times. What is the difference between the theoretical probability of rolling a number less than 4 and the experimental results recorded in the table above? A. 8%C. 58% B. 79%D. 29% OutcomeFrequency 17 212 310 48 58 64
U SING PROBABILITY TO MAKE PREDICTIONS Since history tends to repeat itself, probability is used to make predictions. To make predictions with probability, you set up a proportion. Example: Last basketball season, John made 60% of the free throws he attempted. In his 1 st game this season, he went to the free-throw line 8 times. How many free throws did John make if his success rate from last year continued?
A NSWER TO EXAMPLE PROBLEM John would have made about 5 of his 8 free throws in order for his success rate to continue. Another way to solve… Convert 60% to a decimal number and multiply times 8: 60% =.6 × 8 = 4.8
P RACTICE PROBLEM #7 The student council surveyed a random sample of students and asked which of the following activities the students would prefer as a school trip. There are 2,340 students in the school. Based on the data in the survey, how many students are likely to choose skating? A. 234C. 351 B. 260D. 468
P RACTICE PROBLEM #8 The probability of a table-tennis ball being defective is. About how many balls would be defective in a case of 725 table-tennis balls? A. 1 B. 7 C. 73 D. 80
A DDITIONAL PRACTICE Now, log in to Study Island at www.studyisland.com. www.studyisland.com Click on My Classes in the left-hand side bar. Choose my class, if necessary. Select the assignment titled Probability and work 10 problems in test mode. I will check Study Island to see who completed the assignment and, YES, it is for a grade! When finished w/ Study Island, you may play probability games listed on http://classroom.jc-schools.net/basic/math- prob.html. http://classroom.jc-schools.net/basic/math- prob.html
A NSWER TO PROBLEM #3 Correct answer: A Solution: Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes. Step 2: The probability of the coin lands on the black square is 32. Step 3: Total number of outcomes = 64. Step 4: P (event) =
A NSWER TO P ROBLEM #4 Correct Answer: B Solution: Step 1: Experimental probability = # of times the event occurs ÷ total # of trials Step 2: # of times a head appears = 27. Step 3: Total # of experiments = 60 Step 4: So, the experimental probability of getting a head is…
A NSWER TO PROBLEM #5 Correct Answer: C Six out of 15 rolls resulted in a 1, which simplifies to 2 out of 5.
A NSWER TO PROBLEM #6 Correct answer: A Number times a number less than 4 was rolled in the experiment = 29 Number times a number less than 4 should theoretically be rolled = 25 (1/2 of 50) Difference: 29 – 25 = 4 4 ÷ 50 =.08 = 8%
A NSWER TO PROBLEM #7 Correct answer: A 200 of the 2,340 students were surveyed. Of those 200, 20 would prefer to go skating. There are a couple of ways to solve this… 1) 2) (20 ÷ 200) × 2340 = 234
A NSWER TO PROBLEM #8 Correct answer: H (1/10) × 725 = 72.5