Holt CA Course 1

8-3 Theoretical Probability

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Holt CA Course 1

8-3 Theoretical Probability

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Gina takes a marble from a bag, records the color, and then replaces it. She repeats these steps 20 times. Her results are 6 red marbles, 12 blue marbles, and 2 green marbles. Find the experimental probability of each event.

1. drawing a red marble from the bag

2. drawing a blue marble from the bag

3. drawing a green marble from the bag

35

310

110

Holt CA Course 1

8-3 Theoretical Probability

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.

California Standards

Holt CA Course 1

8-3 Theoretical Probability

Vocabulary

theoretical probability

Holt CA Course 1

8-3 Theoretical Probability

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.

Holt CA Course 1

8-3 Theoretical Probability

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.

Holt CA Course 1

8-3 Theoretical ProbabilityAdditional 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

P = number of ways the event can occur total number of equally likely outcomes

P(clear) = number of clear marblestotal number of marbles

= 0.45 = 45%

Write the ratio.

Substitute.

Write as a decimal and as a percent.

= 920

The theoretical probability of drawing a clear marble is , 0.45, or 45%.920

Holt CA Course 1

8-3 Theoretical ProbabilityAdditional Example 1: Finding Theoretical Probability

P = number of ways the event can occur total number of equally likely outcomes

P(blue) = number of blue marblestotal number of marbles

= 1120

= 0.55 = 55%

The theoretical probability of drawing a blue marble is , 0.55, or 55%.1120

Write the ratio.

Substitute.

Write as a decimal and as a percent.

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

Holt CA Course 1

8-3 Theoretical Probability

P = number of ways the event can occur total number of equally likely outcomes

P(green) = number of green marblestotal number of marbles

= 0.4 = 40%

Write the ratio.

Substitute.

Write as a decimal and as a percent.

Check It Out! Example 1A. 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.

= 820

The theoretical probability of drawing a green marble is , 0.4, or 40%.820

Holt CA Course 1

8-3 Theoretical Probability

P = number of ways the event can occur total number of equally likely outcomes

P( > 4) = 2 numbers greater than 4 6 possible outcomes

= 26

= 13

0.33 33%

The theoretical probability of rolling a number greater than 4 is 0.33, or 33%. 1

3 ,

Check It Out! Example 1

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.

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.

Write the ratio.

Substitute.

Write as a decimal and as a percent.

Holt CA Course 1

8-3 Theoretical Probability

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.

P(boy)= 1323

P(boy) = number of boys on the team

number of members on the team

Holt CA Course 1

8-3 Theoretical Probability

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

Remember!

Holt CA Course 1

8-3 Theoretical ProbabilityAdditional Example 2: School Application

B. Find the theoretical probability of drawing a girl’s name.

+ P(girl) = 11323

P(boy) + P(girl) = 1

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.

Substitute for P(boy). 1323

Subtract from both sides. 1323

– = –1323

1323

P(girl) = 1023 Simplify.

Holt CA Course 1

8-3 Theoretical Probability

Check It Out! Example 2

= 1227

P(girl) = number of girls in the classnumber of students in the class

A. Find the theoretical probability that a girl’s name will be drawn.

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.

Holt CA Course 1

8-3 Theoretical ProbabilityCheck It Out! Example 2

B. Find the theoretical probability that a boy’s name will be drawn.

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.

+ P(boy) = 11227

P(girl) + P(boy) = 1

Substitute for P(girl). 1227

Subtract from both sides. 1227

– = –1227

1227

P(boy) = 1527 Simplify.

Holt CA Course 1

8-3 Theoretical Probability

Lesson QuizFind the probabilities. Write your answer as a ratio, as a decimal to the nearest hundredth, and as a percent to the nearest whole percent. You have 11 cards, each with one of the letters from the word mathematics.

1. Find the probability of drawing an m from the pile of shuffled cards.

2. Find the probability of drawing a vowel. 3. Find the probability of drawing a consonant.

211

, 0.18, 18%

4 11

, 0.36, 36%

711

, 0.64, 64%