Probability Guide Book

1. Probability Guide Book By: A Former Student Accelerated Class

2. Part 1- Probability Probability tells how likely an event is to occur. It can be expressed as either a fraction or a percent. Probability is used in every day life from predicting the weather to finding the chance that you’ll win the lottery! To find the probability of an event happening, divide the number of favorable outcomes that can possibly occur by the total number of possible outcomes. The total number of possible outcomes is called the sample space. Example: Sarah places ten marbles in a bag. There are three striped, six white, and one black marble. The sample space is ten because there are ten possible outcomes.

3. Part 1- Probability Example: Morgan is playing a card game with a normal deck of cards (52 cards). She draws a card randomly from the deck. What is the probability that she will draw an ace? - There are 52 total possible outcomes. - The favorable outcome for this problem is the aces. There is one ace for each of the 4 suites, so there are 4 favorable outcomes. - Write the number of favorable outcomes over the total number of outcomes. - 4/52, simplified to 1/13 is the probability of drawing an ace from the deck.

4. Part 1- Probability Example: Robert rolls two fair number cubes. What is the chance that the sum of the two number cubes will be an even number? Probability of rolling an even on the first cube multiplied by the probability of rolling an even number on the second cube. Answer: ¼

5. Part 2- Experimental Probability Experimental probability is a ratio of the total number of favorable outcomes that occurred in the experiment to the total number of trials. Experimental probability is found by the actual outcomes of an experiment, which differs from theoretical probability, which is found by dividing the number of favorable outcomes that can possibly occur by the total number of possible outcomes. Experimental probability is determined by the results of an experiment, and the probability, though it will be close, will differ slightly from the theoretical probability. Examples: 50 people were surveyed about their favorite ice cream flavors. The following table shows the results. What is the theoretical probability of people chocolate as their favorite flavor? Answer: 16/50 (simplified to 8/25)

6. Part 2- Experimental Probability Example: Jack decided to flip three coins and record the number of times they landed on either all heads, all tails, two heads and one tail, or two tails and one head. The table shows his results. What is the experimental probability that he landed on all heads? The favorable outcome is all heads. The coins landed on all heads twice out of the ten trials. The probability is written with the number of times the favorable outcome occurred over the total number of trials. The experimental probability of landing on all heads is 2/10 simplified to 1/5.

7. Part 3- Theoretical Probability Theoretical probability is the number of favorable outcomes that could possibly occur divided by the number of total possible outcomes. Theoretical probability predicts how likely an event is to occur based on the possible outcomes, unlike experimental probability, which is based on the results of an experiment. Example: Taylor spins a spinner like the one shown. What is the theoretical probability of the spinner landing on a vowel? Outcomes: 8 Favorable outcomes: 3 Answer: 3/8

8. Part 3- Theoretical Probability Sam writes the letters P, R, O, B, A, B, L, I, T, and Y on 10 cards and puts them in a bag he draws one card randomly. (1.) What is the probability of him choosing a card with a B on it, and (2.) what is the probability of him drawing a card with a consonant on it if Y is not counted as a vowel? 1. The favorable outcome is a card with a letter B. There are 2 cards with a B written on it, so 2 will be written over 10 (the number of total possible outcomes. 2/10 can be simplified to 1/5. Answer: 1/5 2. The favorable outcome is a card with a consonant written on it. There are 7 consonant cards in the bag, so 7 will be written over 10 (the total number possible outcomes). Answer: 7/10

9. Part 4- Tree Diagrams & The Counting Principle A tree diagram is a way that we can show all the possible outcomes that are possible for two or more events. Example: Amanda is going on a skiing trip. She can get to the ski hill by either airplane or train and she can pick between either a full or half day ticket. She can stay in either a hotel or a cabin. Make a tree diagram to show all the possible ways that Amanda can spend her ski trip. Solution:Hotel Half day Cabin Airplane Hotel Full day Cabin Hotel Half day Cabin Train Hotel Full day Cabin

10. Part 4- Tree Diagrams & The Counting Principle The Counting Principle is the principle that is used to find the number of possible outcomes of two or more events. Examples: The lock box outside Jason’s house has a 6 digit code. If the first digit is 2, then how many possible number combinations could the code be? There are 10 possible numbers that each digit of the code could be, except for the first digit. Since the first digit is 2, then there is only one possible number that that digit could be. Multiply the number of possible numbers for each of the 6 digits (1 x 10 x 10 x 10 x 10 x 10).There are 100,000 possible codes for the lock box. A painter is painting a house. He asks the owner to pick a color for the front door, the trim, the brick, and the back door. The owner can pick either red, green, or black for each door, either white, purple, blue, green, or gray for the brick, and white, gray, or black for the trim. How many possible ways can the painter paint the house? Answer: 135 ways

11. Part 5- Independent and Dependent Events Two events are independent if the occurrence of one event doesn’t effect the probability of the of the second event taking place. Two events are dependent if the occurrence of one event does effect the probability of the second event occurring. Example of Independent Events: Erica draws a card from a deck of 52 cards, records the answer, then replaces the card in the deck. She then draws another card. What is the probability that both cards will be clubs? Example of Dependent Events: A teacher randomly picks a student from her class of 21 children to be a leader of a group. She then randomly picks another student out of the remaining 20 students to be a second leader of a group. What is the probability that Tomas and Gabriela will be picked?

12. Part 5- Independent and Dependent Events Dependent Events: Beatrice chooses a marble out of a bag of 3 red marbles, 4 black marbles, 1 blue marble, and 2 yellow marbles. She then chooses another marble without replacing the first. What is the probability that she will choose a red marble and then a blue marble? The probability of choosing a red marble is 3/10. Once you have chosen the first marble, there are only 9 possible outcomes left. The probability of choosing a blue marble out of 9 possible outcomes if 1/9. Multiply 3/10 by 1/9. The probability of choosing a red marble and then a blue marble is 3/90, simplified to 1/30. Answer: 1/30 Three friends are sharing a bag of candy. There are 3 red pieces, 2 purple pieces, 4 yellow pieces, 2 green pieces, and 3 white pieces. The first friend reaches into the bag and pulls out 3 pieces of candy. What is the probability of her choosing a red piece and then a yellow piece? Answer: 6/91

13. Part 5- Independent and Dependent Events Independent Events: Elizabeth tells her two friends to each guess a number from one to 15. What is the probability that both her friends guess 10? Answer: 1/225 In a board game, players roll a 6-sided die, spin a spinner, and pick a card from a normal deck of cards (52 cards). There is 4 sections on the spinner: red, green, yellow, and blue. What is the probability that a player will roll a 3, land on red, and pick an ace? There is a 1/6 chance of rolling a 3 on a 6-sided die. There is a ¼ chance of landing on red. There is a 4/52 chance (simplified to 1/13 chance) of picking an ace. To solve the problem, multiply 1/6 by ¼ by 1/13. Answer: 1/312