TOPIC: Experiment and Sample Space

Experiment and Sample Space LESSON 9.1 EXPERIMENTS AND SAMPLE SPACE What You'll Learn Connect People deal with probability 1 Define an experiment, outcome, sample space, and event. questions on a daily basis. Now, 2 List all the possible outcomes in an experiments. you will attempt to predict the chance that something will or will not happen. 9.1.1 THE CONCEPT OF CHANCE Explore Many things in life are uncertain. Will our team win the game? Will it rain Will I pass today? the test? mese questions can be answered with words such as "maybe", "likely", "perhaps", or "probably". We ask such questions because we are uncertain whether such events will occur or not.Consider which of these events is likely to happen. It rains on Christmas Day. It snows in Manila. A storm hits the Philippines in April. Our personal experiences tell us that rain on Christmas Day is highly likely, but a storm in April is quite unlikely. Snow in Manila, on the other hand, is extremely unlikely. People use their personal experiences and intuition to answer questions about an event's chance of occurring, but in some cases, it is possible to give a more precise numerical measure of chance. This is known as probability. Probability is the branch of mathematics concerned with analyzing the chance that a particular event will occur. Its purpose is to attempt to predict the likelihood that something will or will not happen. EXAMPLE 1 Tell whether each event is certain, impossible, or possible. 1. A mother will give birth to twins. 2. Vinegar is sour. 3. You will go sunbathing tonight. ANSWER 1. It is possible. Some mothers have twins. 2. It is certain. 3. It is impossible since there is no sun at night. Try This Tell whether each event is certain, possible, or impossible. 1. A snake will swim in water. 2. A carabao will climb a tree. 3. You will make it to the list of top ten students in your class.Reflect Does achieving your goals seem like an impossible dream? We are the Champions 9.1.2 PROBABILITY EXPERIMENTS AND SAMPLE SPACE When a mother gives birth to a baby, what are the possible outcomes? The baby could be a boy or a girl. When you toss a coin, what are the possible outcomes? The coin may land head or tail. In these examples, the result depends on chance. The table below shows examples of probability experiments and their possible outcomes. Probability Experiment Possible Outcomes Tossing a die PO o oo 18 Ood Flipping a coin head tail Drawing a card 2 3 5 at random from a set of 4 10 6 1 9 5 7 6 7 9 10 cards 8 (1-10) LESSON 9.1 Experiments and Sample Space 409Possible Outcomes Probability Experiment Spinning a wheel Each possible result of a probability experiment is an outcome. The set of all possible outcomes is the sample space of the experiment. The sample space is usually denoted by S and the total number of possible outcomes by n(S). EXAMPLE 2 For each experiment, write the sample space and the total number of possible outcomes. a. Tossing a die b. Spinning the wheel shown, assuming that the arrow will never fall on the lines separating the 3 sectors c. Flipping a coin d. Drawing a card at random from a deck of 10 identical card suits labeled 1 to 10 6 SOLUTIONS: S = {1, 2, 3, 4, 5, 6} n(S) = 6 b. S = {red, yellow, blue} n(S) = 3 C. S = {H, T} n(S) = 2 d. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} n(S) = 10Try This For each experiment, write the sample space and the total number of possible outcomes. Think About This a. Spinning the wheel shown, assuming that the arrow will never fall on the lines separating the 6 sectors. 12 2 How many numbers are there 10 4 from 0 to 7? Many students answer 8 6 this by subtracting b. Drawing a card at random from a deck of 0 from 7. 7-0=7 8 identical card suits labeled 2 to 9. But there are 8 2 numbers from 0 to 7. Thus, to find how many numbers there are from 0 to 7, add 2 1 to the difference of . Tossing a standard octahedral (8-sided) die. 1 7 and 0. bledw / 7-0+1=8 Try it! How many counting numbers are there from 16 to 86? EXAMPLE 3 There are 4 red blocks and 3 blue blocks in a box with numbers written on each block that cannot be felt by the one drawing the block. Write the sample space and the total number of possible outcomes if a block is picked at random. SOLUTION: Since there are 4 red blocks and 3 blue blocks, S = {R,, R2, R3, R,, B,, B2, By] where R represents the outcome of getting a red block and B represents the outcome of getting a blue block. It is incorrect to write S = {R, B] since it would suggest that there is only one red block and one blue block. So , n (S) = 7. Try This There are 2 red balls, 3 green balls, and 1 black ball in a pouch with numbers written on each ball. Write the sample space and the total number of possible outcomes if a ball is picked at random. LESSON 9.1 Experiments and Sample Space 471Exercises Comprehension Check Tell whether each event is certain, impossible, or possible. 1. A dog will lay eggs. 2. If you dip your hand in water, your hand will get wet. 3. A Filipino does not speak Tagalog. 4. June comes after July. 5. A carabao's color is white. 6. The sun will rise tomorrow morning. 7. A man gives birth to a baby. 8. I will pass the test next week. 9. Sugar dissolves in water. 10. A whale will drown at sea. Tick the event that is more likely to happen. 11. You will obtain a college degree. b . You will become an international celebrity. bas asold bes b ace s 12. a. A cellular phone you just bought is defective. thew visdow b. A cellular phone you just bought is not defective. 13. a . The next Philippine president is a woman. Lot odd lyas sorge sigurise b. The next Philippine president is a man. smooduo aldiznog to asdodon 14 a. You will see a whale shark in Sorsogon. us asbold bar break ashift b . You will climb Mt. Everest. 15 a. A person wins the grand prize in lotto. tooduo ads at b. A person is struck by lightning. If you spin the spinner 50 times, which color will be most likely and least likely to occur? 16. most likely : 17. least likely: A bag has 5 red balls, 1 black ball, 3 blue balls, and 8 yellow balls. If you draw a ball at random and put it back into the bag 20 times, which color will be most likely and least likely to occur? 18. most likely: 19. least likely: 412 CHAPTER 9 Probabilityfor each experiment, write the sample space and the total number of possible outcomes. 20. To decide on which day to visit his grandmother, Bobby writes the 7 days of the week on equal pieces of paper and picks one at random. a. S b. n(S) = 21. Minie tosses a standard tetrahedron die marked 1-4. a. S = 2 b. n(S) = 22. Marco writes down the prime numbers less than 20 on identical pieces of paper, one number on each paper. Then he draws one paper at random. a. S = b. n (S ) = 8 Jessica spins the wheel shown assuming that the arrow will never fall on the 7 2 lines separating the sectors. 6 3 a. S 15 4 b. n(S) = 24. Identify events that are certain, impossible, or possible. Write three examples for each. 1 Impossible N 3. 1 Possible 2. 3 1. Certain 2. 3 413 LESSON 9.1 Experiments and Sample Space