Relative Frequency

RELATIVE FREQUENCY

INTRODUCTION

The objective for this lesson on Relative Frequency is, the student will use data collected from probability to predict relative frequency.

The skills students should have in order to help them in this lesson include, probability and fraction-decimal-percent equivalence.

We will have three essential questions that will be guiding our lesson. Number one, how can we define relative frequency? Number two, if a coin is tossed twenty times and lands on the head of the coin eight times, what is the relative frequency? Number three, explain how the probability of an event differs from the relative frequency.

Begin by completing the warm-up on probability to prepare for the lesson on Relative Frequency.

SOLVE PROBLEM - INTRODUCTION

The SOLVE problem for this lesson is, Jason is going to conduct a probability experiment for his math project. He can use either a fair number cube, a bag with number tiles or a coin toss. He decides to use a coin for his experiment. The first question he needs to answer for his project is about the probability of certain events. What is the probability that Jason will toss the coin and have it land on heads?

S, Study the Problem. Underline the question. This problem is asking me to find the probability that the coin toss will land on heads.

O, Organize the Facts. First we identify the facts. We read the problem again and make a vertical strike mark after each fact. Jason is going to conduct a probability experiment for his math project./ He can use either a fair number cube, a bag with number tiles or a coin toss./ He decides to use a coin for his experiment./ The first question he needs to answer for his project is about the probability of certain events./

Then we eliminate the unnecessary facts. A fact is only necessary if it helps answer our question, what is the probability that Jason will toss the coin and have it land on heads?

Then we list the necessary facts. Toss a coin, two sides – heads or tails

L, Line Up a Plan. Write in words what your plan of action will be. Create a ratio that shows the relationship between the desired outcome heads and the number of possible outcomes. Convert the ratio to a decimal and a percent.

Choose an operation or operations. Division

V, Verify Your Plan with Action. First estimate your answer. And we have an estimate here of one half.

Then carry out your plan. We create the ratio that shows the relationship between the desired outcome of heads and the number of possible outcomes. Our answer is one half, which is equivalent to zero point five, which is equivalent to fifty percent.

E, Examine Your Results

Does your answer make sense? Compare your answer to the question. Yes, because we are looking for the probability that the coin toss will land on heads.

Is your answer reasonable? Compare your answer to the estimate. Yes, because it matches our estimate of one half.

Is your answer accurate? Check your work. Yes

Write your answer in a complete sentence. The probability that the coin toss will land on heads is one half, which is equivalent to zero point five, which is equivalent to fifty percent.

DISCOVERY ACTIVITY – EXTEND THE SOLVE PROBLEM – PROBABILITY AND RELATIVE FREQUENCY

Take a look at the questions below the SOLVE problem.

How many possibilities were there for the coin toss? Two

What was the probability of tossing the coin and having it land on heads? One half, which was equivalent to zero point five, which was equivalent to fifty percent.

Do you think that the coin will always land on heads every one out of two times? No. Why not? Depending on how the coin is tossed, how it lands or where it lands can alter the outcome of the coin flip.

Jason began working on his math project by tossing the coin twenty times. The results of his experiment are shown in the table.

What is the probability of tossing a heads on the coin? Ten out of twenty, which is equivalent to five tenths or zero point five, which is equivalent to fifty percent.

How did Jason determine this? Half of the tosses should be heads. This would be ten out of twenty, which equals zero point five or fifty percent.

What outcome was Jason hoping for when he tossed the coin? Heads

How many coin tosses were heads? Twelve

When we talk about how often or how many times something happens, we talk about frequency of that event. For example, if Jason’s math teacher gives homework assignments three days every week, the frequency of having homework is three times per week.

What was the result of Jason’s experiment? The coin landed on heads twelve out of twenty tosses. This means the frequency, or how often the coin landed on heads, was twelve.

If you want to compare the relationship between the frequency of tossing heads on the coin and how many times the coin was tossed, we need to know the relationship between the two numbers.

What were the three forms we used to write the relationship? Ratio, decimal or percent

When you describe how you are related to someone in your family, you that say he or she is your relative.

When we described the relationship between how often Jason tossed a head and the total number of events, we described how one value was related to the other.

When we combine the number of times the event occurred, frequency with the description of how the two values are related, relative, we know the relative frequency of Jason tossing a head on the coin.

Create a definition of relative frequency. The ratio or relationship between the number of favorable events to the total number of events is the relative frequency.

Describe the comparison of the probability and the relative frequency of Jason’s project. The relative frequency was greater than the probability.

Complete the coin toss experiment for thirty and then forty tosses.

How did the probability for thirty coin tosses compare to the relative frequency?

How did the probability for forty coin tosses compare to the relative frequency?

Based on the experiments for thirty and forty coin tosses, what do you predict would happen with fifty coin tosses? The relative frequency would be closer to the probability.

Based on the experiments for thirty and forty coin tosses, what do you predict would happen with one hundred coin tosses? The relative frequency would be closer to the probability.

RELATIVE FREQUENCY AND PROBABILITY WITH CARDS

Take a look at the top of the page with the large graphic organizer.

Use the set of cards given to you by your teacher, take out a two, three, four, five and six.

What is the probability of choosing a two? One fifth or one out of five

Explain the probability of two. There is only one card with a two. There are a total of five cards, so the probability is one out of five, which equals zero point two, which equals twenty percent.

What is the probability of choosing a three? One over five, one fifth

Explain the probability of three. There is only one card with a three. There are a total of five cards, so the probability is one over five, which is equivalent to zero point two, which is equivalent to twenty percent.

What is the probability of choosing a four? One over five, or one fifth

Explain the probability of four. There is only one card with a four. There are a total of five cards, so the probability is one fifth or one over five, which is equivalent to zero point two, which is equivalent to twenty percent.

What is the probability of choosing a five? One fifth, or one over five

Explain the probability of five. There is only one card with a five. There are a total of five cards, so the probability is one over five, which is equivalent to zero point two, which is equivalent to twenty percent.

What is the probability of choosing a six? One fifth or one over five

Explain the probability of six. There is only one card with a six. There are a total of five cards, so the probability is one over five, which is equivalent to zero point two, which is equivalent to twenty percent.

Take five cards two through six from the number cards and place them face down on your desk. Complete ten trials for each of the five probabilities and mark the number of times that a number occurred in the chart. Then, fill in the relative frequency column. Record your answer each time.

How did the probability differ from the relative frequency?

Was the relative frequency greater than the probability?

Was the relative frequency less than the probability?

Were they the same?

Discuss why the results may have happened. With the relative frequency there are variables such as choosing the card. There were only ten trials.

What other reasons might there be for the results?

Did each student have the same relative frequency?

While answers may vary, most likely, student had different results because of the differences in the experimental process.

Discuss the meaning of the following terms and write their definitions.

Probability: How likely something is to happen.

Relative Frequency: The observed number of successful events or outcomes for a number of trials.

CONCLUSIONS ABOUT RELATIVE FREQUENCY AND PROBABILITY

Take a look at the graphic organizer on the next page.

Identify the event in Situation One. A coin toss two hundred times

What was the outcome of the coin toss? One hundred eight heads, and ninety two tails

What is the probability of flipping a head on the coin? One hundred out of two hundred, which is equivalent to one half, which is equivalent to fifty percent

What was the relative frequency of heads in this situation? One hundred eight out of two hundred, which is equal to fifty four percent

Identify the event in Situation Two. A coin toss one hundred twenty times

What was the outcome of the coin toss? Fifty heads, seventy tails

What is the probability of flipping a tail on the coin? Sixty out of one hundred twenty, which is equal to one half, which is equal to fifty percent

What was the relative frequency of the tails in this situation? Seventy out of one hundred twenty, which is equivalent to fifty eight point thirty three percent

Which situation had a relative frequency that was closer to the probability of fifty percent? The coin toss with two hundred trials

Can you make a prediction why that was the result? The situation with more trials has a smaller difference between the probability and the relative frequency.

Based on this information, what do you predict would happen if the experiment had five hundred trials? The difference between the probability and the relative frequency would be smaller than the situation with one hundred trials and closer to fifty percent.

PROBABILITY FOLDABLE

Step One: Label the second flap “Relative Frequency.” On the inside complete the section for Relative Frequency with the given information.

CLOSURE

Now let’s go back and discuss the essential questions from the lesson.

Our first question was, how can we define relative frequency? Relative Frequency is observed number of successful events or outcomes for a number of trials.

Number two, if a coin is tossed twenty times and lands on the head of the coin eight times, what is the relative frequency? The relative frequency is eight out of twenty, which is written as the ratio eight over twenty, which is equal to zero point four, which is equal to forty percent.

And number three, explain how the probability of an event differs from the relative frequency. The probability is a numerically established value. When determining the relative frequency, the outcomes will vary with the sample size. The larger the same size, the closer the relative frequency will be to the probability.