Game of Chance

Getting Started

The student does not understand how to use probability to determine an expected frequency.

The student:

• Restates the probability as a fraction or percent.
• Attempts to find a frequency based on an assumption about the number of marbles in the bag.
  
  
  
  
• Creates a proportion that does not correctly represent the situation.
• Provides an answer without work or explanation.

The probability of drawing a green marble is 0.6. What does that mean?

What is the question asking you to find?

How do you know how many marbles there are in the bag? Do you need to know this to answer the question?

Be sure the student understands the scenario and the concept of replacement. Explain that, because Dylan replaces the marble after each draw, the probability of drawing a green marble is the same for each draw. Ask the student to hypothesize whether or not it is necessary to know the number of marbles in the bag if the probability is given.

Explain to the student that the frequency of the occurrence of an event can be estimated based on a known probability (either theoretical or experimental). Guide the student to understand that if one expects to select a green marble 0.6 or 60% of the time, then over 50 draws, one would expect to select a green marble 60% of 50 (or 0.6 x 50 = 30) times. Clarify that it is not necessary to know how many marbles are in the bag. But, explain that it was very important to know this when originally determining the probability of selecting a green marble from the bag. Ask the student to estimate the number of times one would expect to draw a purple marble in 200 draws.

Provide additional opportunities to estimate the frequency of an event based on a given probability.

Consider implementing other MFAS tasks for this standard.

Making Progress

The student does not understand the relationship between expected outcomes and long-run relative frequency.

The student says that he or she would expect 30 green marbles to be drawn out of 50 tries based on the calculation 0.6(50) = 30. When asked if it is possible that Dylan could draw exactly five green marbles in 50 trials, the student:

• Says it is not possible because 30 green marbles will be drawn.
• Says there are not enough green or too many green marbles for this to happen.
  
  
  
  
• Refers to the number of marbles in the bag.
  
  

Do you think it is possible that what happens in reality could differ from a prediction based on a probability?

If the probability of getting a green marble is , does that mean that you will always get six green marbles out of ten tries?

Use manipulatives to demonstrate the relationship between probabilities, expected frequencies, and observed frequencies (e.g., the probability of getting “heads” when flipping a coin is ). Ask the student to flip a coin two times and compare his or her results to the expected frequency ( x 2 = 1). Discuss how the small number of trials might account for any differences in the expected and observed frequencies. Explain probability in terms of what is expected to occur in the long run.

Conduct a simulation of the coin flipping experiment. Note the observed frequency of heads after 1, 2, 3, 4, 5, 10, 50, 100, and 1000 trials. Guide the student to observe that in the long run, the number of heads converges on the expected frequency.

Got It

The student provides complete and correct responses to all components of the task.

The student says that he or she would expect 30 green marbles to be drawn out of 50 tries based on the calculation 0.6(50) = 30. The student says that it is possible for Dylan to draw exactly five green marbles in 50 tries:

• Because 30 is only an estimate.
  
  
  
  
• Even though it is not very likely.
  
  

Is it possible for Dylan to draw no green marbles at all?

What is the probability that Dylan will draw a purple marble on any given draw?

Provide additional opportunities to explore the difference between expected frequencies and observed frequencies. Ask the student to determine the difference between observed frequencies and expected frequencies in four trials, 10 trials, or 50 trials. Guide the student to observe what is happening in the long run.

Consider implementing other MFAS tasks for this standard.