Getting Started 
Misconception/Error The student does not understand the meaning of numerical representations of a probability. 
Examples of Student Work at this Level The student shows a lack of clarity about the likelihood of an event. The student:
 Bases decisions of likelihood on something other than its probability. For example, the student says a probability:
 Cannot be zero because â€śzero in not on a number cube.â€ť
 Cannot be zero or one because they are â€śnot a fraction or percent.â€ť
 Depends on what the event is about.
 Misinterprets probabilities given as fractions, saying you may or may not â€śgetâ€ť the numerator value. For example, the student says:Â
 You might not get the one in or .
 Youâ€™re most likely to get the nine in .
 Interprets one as â€śnot likelyâ€ť because:
 It is â€śa very low numberâ€ť or â€śclose to zero.â€ť
 It is equivalent to one out of 100 or 1%.
 Reverses the meaning of the likelihood of numbers close to one and zero.Â

Questions Eliciting Thinking What kinds of numbers can represent probabilities? What is the largest a probability can be? What is the smallest a probability can be?
When a probability is given as a fraction, what do the numerator and denominator mean?
How is a probability of one different from a probability of ?
What if something could never happen? What number describes its probability? 
Instructional Implications Help the student understand that the probability of an outcome or event is a number between zero and one. Larger numbers indicate greater likelihood. A probability near zero indicates an unlikely event, a probability around indicates an event that is equally likely as unlikely, and a probability near one indicates a likely event. Assist the student in assigning numerical probabilities to realworld events that are easily understood. Describe an event that could never occur (e.g., the probability of snow in Miami in July), that is certain to occur (the probability that the sun will rise tomorrow), and that is as likely to occur as not (e.g., the probability the next person to walk into the room is a male). Assign numerical probabilities to these events to assist the student in developing an intuitive understanding of numerical representations of probability. Ask the student to describe events with given probabilities and to assign numerical probabilities to events that are described.
Provide the student with opportunities to explore and calculate the probabilities of outcomes that vary in terms of the likelihood of their occurring. For example, using marbles of two different colors (e.g., red and blue), show the student a set of 10 marbles that includes one red marble and nine blue marbles. Then ask the student to describe (in words) how likely it would be to select a red marble and how likely it would be to select a blue marble if a marble were randomly selected from a bag containing the 10 marbles. After verbally describing the probability of selecting a marble of each color, calculate the theoretical numerical probabilities and relate the numerical probabilities to the verbal descriptions. Repeat this exercise using sets that contain: (1) no red marbles and 10 blue marbles; and (2) five red marbles and five blue marbles. 
Making Progress 
Misconception/Error The student is not precise in explanations of probabilities. 
Examples of Student Work at this Level The student understands that probabilities range from zero to one but does not use precise wording to clarify variations in likelihood. For example, the student:
 Does not use words like â€ścertainâ€ť or â€śimpossibleâ€ť for probabilities of one and zero, respectively.
 Does not use words like â€śvery likelyâ€ť or â€śvery unlikelyâ€ť for probabilities of and , respectively.
 Does not explain as â€śequally likely as unlikelyâ€ť using instead only â€ślikelyâ€ť or â€śunlikely.â€ť
 Does not distinguish between small but significant differences in probabilities, writing that means, â€śItâ€™s not going to happen,â€ť and that means, â€śIâ€™m sure it will happen.â€ť

Questions Eliciting Thinking What words could you use to describe the probability of zero and one other than â€ślikelyâ€ť and â€śunlikelyâ€ť? How likely or unlikely are they?
You described as likely? How likely is it? Would it be correct to also describe it as unlikely?
What makes you think that an event with a probability of â€śis not going to happenâ€ť? Is there any chance of it happening? What is a better way to describe it? 
Instructional Implications Assist the student in using language that more precisely defines probabilities. Model explaining that a probability of, for example, indicates an event is very unlikely but could occur. Provide the student with opportunities to explore and calculate the probabilities of outcomes that vary in terms of the likelihood of their occurring. For example, using marbles of two different colors (e.g., red and blue), show the student a set of 10 marbles that includes one red marble and nine blue marbles. Then ask the student to describe (in words) how likely it would be to select a red marble and how likely it would be to select a blue marble if a marble were randomly selected from a bag containing the 10 marbles. After verbally describing the probability of selecting a marble of each color, calculate the theoretical numerical probabilities and relate the numerical probabilities to the verbal descriptions. Repeat this exercise using sets that contain: (1) no red marbles and 10 blue marbles; and (2) five red marbles and five blue marbles.
Consider implementing the MFAS tasks Probability or Not? and Likely or Unlikely?Â to further assess understanding of the range of possibilities for the probability of outcomes and events. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student gives correct descriptions of each probability. For example, the student responds:
 1: â€ścertainâ€ť or â€śfor sure will happenâ€ť
 :Â â€śvery unlikelyâ€ť or â€śmost likely will not happenâ€ť
 0: â€śimpossibleâ€ť or â€śit cannot happenâ€ť
 :Â â€śas likely to happen as not happenâ€ť or 'neither likely nor unlikely'
 :Â â€śvery likelyâ€ť or â€śalmost for sure will happenâ€ť

Questions Eliciting Thinking Is it possible that something with a probability of actually happens 50 out of 100 times?
Can you give an example of an event for each of these probabilities?
Is it ever possible for a probability to be greater than one or less than zero? 
Instructional Implications Consider implementing the MFAS tasks:Â Hens Eggs, Game of Chance, or Probabilities CubedÂ to assess finding the probability of various outcomes and events. 