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# Probability Topic

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PROBABILITY What is probability? Probability is the possibility that a certain event will occur. 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. Probability is the chance of an event occurring divided by the total number of possible outcomes. Different types of events will have their probabilities figured out differently. The probability of 1 single event occurring is figured out differently than the probability of 2 events occurring. Probability is also based on whether events are dependent or independent of each other. To figure out probability, a sample space is used. A sample space lists all the possible outcomes for a certain event. Another tool used in probability is the tree diagram. The tree diagram shows the different combinations that can occur. The counting principle also is used to show different combinations. How to use probability: • The probability of one event occurring is equal to the chance of the event occurring divided by the total outcomes. For example, the probability of picking a seven out of a standard deck of cards is 4/52, or 1/13. • If the probability of picking two events needs to be calculated, the probability would be equal to the probability of the one event plus the probability of the second event. For example, the probability of picking a seven or a jack out of a standard deck of cards would be 4/52 + 4/52 = 8/52, or 2/13. These are called 'or' probability. © 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.

• If the events overlap, it is called 'and' probability. This occurs when asked to find the probability of picking a seven and a heart from a standard deck of cards. The probability of picking a seven is 4/52, the probability of picking a heart is 13/52, but since there is a seven of hearts 1/52 must be subtracted. Example: 4/52 + 13/52 - 1/52 = 16/52 or 4/13 • This could also be shown using a sample space. A sample space shows all the possible outcomes for an event. If a spinner, with the letters A-D equally spaced, is spun and a die is rolled, the probability of getting a 2 and a B could be found by using a sample space. Sample space { A1, A2, A3, A4, A5, A6, B1, B2, B3, B4, B5, B6 } { C1, C2, C3, C4, C5, C6, D1, D2, D3, D4, D5, D6 } The probability of getting a 2 and a B would be 1/24. • A tree diagram is also a visual that shows the combinations of certain events. For example, there are three types of cookies (chocolate chip, oatmeal and sugar) and two types of drinks (milk and juice) to choose from for a snack. How many different combinations are there? Use the tree diagram below: © 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.

This tree diagram shows that there are 6 different ways to have a snack. • This could also be figured out using the counting principle. With the counting principle, the number of different choices is multiplied to get the different combinations. For the above example, 3 cookies x 2 drinks = 6 combinations. The probability of picking sugar cookies and milk is 1/6. • Probabilities can also be found of events that are independent or dependent of each other. Example: If there are 10 marbles in a bag with 4 blue and 6 red, the probability of picking 2 red marbles with replacement 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. Example: The probability of picking 2 red marbles without replacement is 6/10 ·5/9 = 30/90 or 1/3. This probability is dependent because the events are related to each other. © 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.

Try this! 1. Find the following probabilities: Picking a red card Picking an Ace Picking an Ace or a red card Picking a spade or 6 2. Write a sample space for flipping a coin and rolling a die. 3. Draw a tree diagram for 3 shirts (white, black, red) and 2 pants (blue, black). 4. Use the counting principle for 4 main meals, 3 drinks and 3 desserts. 5. In a bag there are 7 yellow marbles and 8 blue marbles. a. Find the probability of picking 1 yellow marble and 1 blue marble with replacement. b. Find the probability of picking 1 yellow marble and 1 blue marble without replacement. © 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.