Theoretical probability and counting
Mathematics, Grade 8
Theoretical probability and counting
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Theoretical probability and counting
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Study Guide Theoretical probability and counting Mathematics, Grade 8
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THEORETICAL PROBABILITY AND COUNTING Theoretical probability is the probability that a certain outcome will occur based on all the possible outcomes. An event that is certain to occur has a probability of 1. An event that cannot occur has a probability of 0. Therefore, the probability of an event occurring is always between 0 and 1. • The closer a probability is to 1, the more certain that an event will occur. • Theoretical probability is the chance of an event occurring divided by the total number of possible outcomes. • Just as probability refers to the possibility of an event happening, odds refer to the odds against an event happening. The Counting Principle is used to find the different combinations or ways two or more events can happen. • To find the number of different ways that two or more events can happen, one event is multiplied by another event. • If there are more than two events, all the events are multiplied together to find the total number of ways those events can happen. • Sometimes, the number of ways that an event can happen depends on the order. o A permutation is an arrangement of objects in which order matters. o A combination is a set of objects in which order does not matter. • Probability is also based on whether events are dependent or independent of each other. o An independent event refers to the outcome of one event not affecting the outcome of another event. o A dependent event is when the outcome of one event does affect the outcome of the other event. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
How to use theoretical probability and counting Theoretical probability is the probability that a certain outcome will occur based on all the possible outcomes. • For example, the probability of picking a 3 out of the numbers 1 - 10 is 1/10. Even if the numbers were picked 10 times, the probability would be 10/100 or 1/10. Since probability is divided by total outcomes, it is useful to be able to figure out the total outcomes. With the Counting Principle, the number of different event choices is multiplied to get the different outcomes. • For example, how many outcomes can there be for a person to pick a snack from 3 cookies and 2 drinks? With the counting principle, there are 3 · 2 or 6 outcomes. Permutations are another way to find the number of outcomes, but in the case of a permutation, order matters. • For example if there are 3 students and 3 chairs, the way the can be arranged is as follows: Ex. There are 3 chairs, ___, ___, ___. There are 3 students to choose from for the first chair, _3_, ___, ___. There are 2 students to choose from for the second chair, _3_, _2_, ___. There is one student left for the last chair,_3_, _2_, _1_. The number of ways the 3 students can be arranged in 3 chairs is 3 · 2 · 1 or 3! = 6 ways. The notation, a!, means factorial, which is the product of the consecutive numbers from a to 1. A combination is another way to figure out total outcomes, but in the case of combinations, order does not matter. • For example, how many different combinations can be made when picking 2 letters out of the word DOG? There are 3 outcomes, DO, DG and OG. Since order does not matter, DO and OD are considered the same. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
Probabilities of independent or dependent events are based on how one event affects the other event, if at all. • For example, if there are 10 marbles in a bag with 4 blue and 6 red marbles, the probability of picking a red marble, putting it back and then picking another red marble is 6/10 · 6/10 = 36/100 or 9/25. This probability is independent because what happened the first time does not affect what happens the second time. The probability of picking a red marble and then another red marble without replacing the first is 6/10 ·5/9 = 30/90 or 1/3. This probability is dependent because the first event affects the second event. Another concept with probability is odds. Odds refer to the odds against an event happening. Odds are used to compare unfavorable possibilities with favorable possibilities. Try This! 1. What is the probability of picking a black card out of a deck of 52 cards? 2. Using the Counting Principle, how many outcomes are there for a pizza with 2 types of sauces, 3 types of cheeses and 8 types of toppings? 3. How many permutations can be made by arranging the letters in the word, MATH? 4. How many combinations can be made from picking two flavors of ice cream out of the flavors, strawberry, vanilla, chocolate, and mint? 5. What is the probability of picking a red card out of a deck of 52 cards, replacing it and then picking out an ace? 6. There are 4 red, 6 yellow and 5 blue marbles in a bag. What is the probability of picking a red marble, and without replacing it, then picking out a blue marble? 7. What are the odds against rolling the number 5 on a die? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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