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California Standards

Learn how to calculate the theoretical probability of drawing marbles from a bag and express the results as ratios, decimals, and percentages.

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California Standards

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  1. California Standards SDAP3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1 – P is the probability of an event not occurring.

  2. Vocabulary theoretical probability

  3. In a board game, players use tiles with the letters of the alphabet to form words. Of the 125 tiles used in the game, 15 have the letter E on them. To determine the probability of drawing an E, you can draw tiles from a bag and record your results to find the experimental probability, or you can calculate the theoretical probability. Theoretical probability is used to find the probability of an event when all the outcomes are equally likely.

  4. If each possible outcome of an experiment is equally likely, then the experiment is said to be fair. Experiments involving number cubes and coins are usually assumed to be fair.

  5. 9 20 = The theoretical probability of drawing a clear marble is , 0.45, or 45%. 9 20 Additional Example 1: Finding Theoretical Probability Andy has 20 marbles in a bag. Of these, 9 are clear and 11 are blue. Find the probability of each event. Write your answer as a ratio, a decimal, and a percent. A. drawing a clear marble number of ways the event can occur total number of equally likely outcomes P = number of clear marbles total number of marbles P(clear) = Write the ratio. Substitute. Write as a decimal and as a percent. = 0.45 = 45%

  6. 11 20 = Additional Example 1: Finding Theoretical Probability Andy has 20 marbles in a bag. Of these, 9 are clear and 11 are blue. Find the probability of each event. Write your answer as a ratio, a decimal, and a percent. B. drawing a blue marble number of ways the event can occur total number of equally likely outcomes P = number of blue marbles total number of marbles P(blue) = Write the ratio. Substitute. Write as a decimal and as a percent. = 0.55 = 55% The theoretical probability of drawing a blue marble is , 0.55, or 55%. 11 20

  7. 8 20 8 20 = The theoretical probability of drawing a green marble is , 0.4, or 40%. Check It Out! Example 1 A. Jane has 20 marbles in a bag. Of these 8 are green. Find the probability drawing a green marble from the bag. Write your answer as a ratio, a decimal, and a percent. number of ways the event can occur total number of equally likely outcomes P = number of green marbles total number of marbles P(green) = Write the ratio. Substitute. Write as a decimal and as a percent. = 0.4 = 40%

  8. number of ways the event can occur total number of equally likely outcomes P = 2 numbers greater than 4 6 possible outcomes P( > 4)= 2 6 1 3 = = The theoretical probability of rolling a number greater than 4 is 0.33, or 33%. 1 3 , Check It Out! Example 1 B. Find the probability of rolling a number greater than 4 on a number cube. Write your answer as a ratio, a decimal, and a percent. For a fair number cube, each of the six possible outcomes is equally likely. There are 2 ways to roll a number greater than 4: 5 or 6. Write the ratio. Substitute. Write as a decimal and as a percent.  0.33 33%

  9. number of boys on the team number of members on the team P(boy) = Additional Example 2: School Application There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. A. Find the theoretical probability of drawing a boy’s name. 13 23 P(boy)=

  10. Remember! The sum of the probabilities of an event and its complement is 1.

  11. 13 23 13 23 Substitute for P(boy). Subtract from both sides. Simplify. Additional Example 2: School Application There are 13 boys and 10 girls on the track team. The name of each of the team members is written on an index card. A card is drawn at random to choose a student to run a sprint and the card is replaced in the stack. B. Find the theoretical probability of drawing a girl’s name. P(boy) + P(girl) = 1 13 23 + P(girl) = 1 13 23 13 23 – = – 10 23 P(girl) =

  12. number of girls in the class number of students in the class P(girl) = Check It Out! Example 2 There are 15 boys and 12 girls in a class. A teacher has written the name of each student on a piece of paper and randomly draws a paper to determine which student will present the answer to the problem of the day. A. Find the theoretical probability that a girl’s name will be drawn. 12 27 =

  13. 12 27 12 27 Substitute for P(girl). Subtract from both sides. Simplify. Check It Out! Example 2 There are 15 boys and 12 girls in a class. A teacher has written the name of each student on a piece of paper and randomly draws a paper to determine which student will present the answer to the problem of the day. B. Find the theoretical probability that a boy’s name will be drawn. P(girl) + P(boy) = 1 12 27 + P(boy) = 1 12 27 12 27 – = – 15 27 P(boy) =

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