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IB Studies www.ibmaths.com Adrian Sparrow Probability 1: the idea of probability.

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1 IB Studies www.ibmaths.com Adrian Sparrow Probability 1: the idea of probability

2 How to write a probability Probabilities are written as fractions or decimals. A probability will never exceed 1 or be less than 0. A probability close to 1 is very likely, if it is 1 it will be certain. A probability close to 0 is very unlikely, if it is 0 it will be impossible. A probability of 0.5, or a half has an even chance.

3 Outcomes Outcomes are all the things that could happen. For example when a normal fair die is rolled there are 6 outcomes: 1,2,3,4,5,6. When a coin is tossed there are 2 outcomes: head or tail. Questions 1. 10 different sweets are in a bag. If one is chosen, how many outcomes are there? 2. There are 15 students in a class. One is chosen, how many outcomes are there? 3. 3 red counters, 2 blue counters and 1 white counter are in a bag. One is chosen. How many outcomes are there? 10 15 6

4 Favourable outcomes and expressing probabilities Favourable outcomes are the ones that are required, usually from the question. Examples 1. A fair normal die is rolled, find the probability that a 5 occurs. 2. 4 red counters, 3 blue counters and 1 white counter are in a bag. One counter is chosen at random. Find the probability of getting a blue counter.

5 Question set 1 1. A fair normal die is rolled. Find the probability that, a) a 4 occurs, b) an even numbers occurs, c) an square number occurs, d) a prime number occurs, e) a 7 occurs. 2. A class of 20 students has 5 Americans, 10 Egyptians, 3 Canadians, and 2 French students. A student is chosen at random. Find the probability of the student being, a) American, b) French, c) Canadian, d) British.

6 Probabilities from observations Probabilities are not always theoretical, as the previous slides. They may be based on some surveys. For example: A student does a survey to find the colour of 50 cars parked in a staff car park. The results are shown below. ColourSilverBlueBlackOther Frequency2012135 Another car is driven into the car park. Based on the results of the survey, find the probability that the car will be, a) blue, b) black, c) a colour other than silver, blue or black.

7 Complementary probabilities All probabilities must add up to 1. 4 red counters, 3 blue counters and 1 white counter are in a bag. One counter is chosen at random. a) Find the probability of getting a blue counter. b) Find the probability of not getting a blue counter. Questions 1. From a group of students it is known that the probability of choosing a female is 0.3. Find the probability of choosing a male.

8 A sample space - more than 1 event Two dice are rolled and the scores are added. How many outcomes are there? The best way to show all the outcomes is to draw a table. 123456 1234567 2345678 3456789 45678910 56789 11 6789101112 This is the main body of the table, showing the 36 outcomes. When two dice are rolled and the scores are added, find the probability of getting, a) a 2, b) a 7, c) a 10, d) a 1.

9 Question set 2 Two dice are rolled and the difference of the numbers is found. a) How many outcomes are there? b) Produce a sample space (table) to list all the outcomes. 123456 1012345 2101234 3210123 4321012 5432101 6543210 Use your table to find the probability of getting a score of, d) a prime number, c) 3, e) a non-prime number, f) not a 3. 0 and 1 are not prime.

10 Expectation Expectation can sometimes be asked. This is how many times we may expect in theory something to happen, and can be found by using probability. Examples If a die is rolled 6 times, how many times would you expect a 5 to occur? If a die is rolled 12 times, how many times would you expect a 5 to occur? If a die is rolled 36 times, how many times would you expect a 5 to occur? If a die is rolled 720 times, how many times would you expect a 5 to occur? You can use probability to answer this: If a die is rolled 324 times, how many times would you expect a 5 to occur?

11 Question set 3


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