MA.8.DP.2.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• NA

Terms from the K-12 Glossary

• Event
• Theoretical Probability

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2.3

Next Benchmarks

• MA.912.DP.4.2
• MA.912.DP.4.3
• MA.912.DP.4.7
• MA.912.DP.4.8
• MA.912.DP.4.9

Purpose and Instructional Strategies

• Instruction builds on finding sample spaces from MA.8.DP.2.1. Have students discuss their understanding of the words “theoretical” and “probability” to build toward a formal definition of theoretical probability.
• Encourage students to use a variety of representations for the sample space, such as a table, tree diagram or list, to assist in determining the total possible outcomes when calculating the probability. Providing opportunities for students to match situations and sample spaces will assist with building their ability to visualize the sample space for any given experiment.
• When finding theoretical probability, have students work from their sample space. Doing so will lead to the understanding that since experiments for this benchmark are fair, the probability of an event is equivalent to .
  • For example, if tossing a fair coin three times, the sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. If one wants to find $P$($a$$t$ $$$l$$e$$a$$s$$t$$$ 1 $$$t$$a$$i$$l$$s$), students can circle all of the outcomes in the sample space that have at least one T. Since there are 7 such outcomes, one can determine the probability as $\frac{\text{7}}{\text{8}}$, or 87.5%.
• Instruction focuses on the experiments listed in Clarification 2.
  • For example, when rolling a 6-sided die twice, $P$($rolling\mathrm{ a}\mathrm{ s}um\mathrm{ o}f$ 7) = $\frac{\text{6}}{\text{36}}$. This can be determined by looking at the table of outcomes and circling the 6 outcomes that give a sum of 7.
  • For example, when picking a card twice with replacement from a deck that contains each of the five vowels of the alphabet (A, E, I, O and U), $P$($n$$o$$t$ $$$p$$i$$c$$k$$i$$n$$g$ $$$a$$n$ $A$) = 0.64. This can be determined by reasoning that there are 9 ways to draw an A, so there are 16 ways to not draw an A.
  • For example, when spinning a spinner twice that contains 3 sections where two of the sections are red and the other section is blue, 
    • P (spinning the same color twice) = $\frac{\text{5}}{\text{9}}$. This can be determined by looking at the list {$r$₁$r$₂, $r$₁b, $r$₂$r$₁, $r$₂b, b$r$₁, b$r$₂, bb, $r$₁$r$₁, $r$₂$r$₂} and circling each of the four outcomes that has the same color twice.

• Instruction includes discussing student understanding of the words “theoretical” and “probability” to develop a formal definition of theoretical probability.
• Instruction includes $P$($e$$v$$e$$n$$t$) notation.
• Students should develop the understanding that the order in which the outcome (from the simple experiment) occurs matters so that probabilities of the outcomes (from the repeated experiment) are the same.
  • For example, if the simple experiment is to draw a marble out of bag (that contains 1 blue, 1 green and 1 yellow marble), the outcomes for that simple experiment are {B, G, Y}. If this experiment is repeated two times, there are now nine outcomes: {BB, BG, BY, GG, GB, GY, YY, YB, YG}. If one wanted the $P$($d$$r$$a$$w$ $$$a$$t$ $$$l$$e$$a$$s$$t$ 1 $$$y$$e$$l$$l$$o$$w$ $$$m$$a$$r$$b$$l$$e$), students should be able to see that the drawing a yellow and then a green is as equally likely as drawing a green and then a yellow. Therefore those are two distinct outcomes.

Common Misconceptions or Errors

• Students may incorrectly assume that all events are equally likely. To help address this misconception, reinforce that the likelihood of each event depends on the number of outcomes in the event.
• Students may incorrectly convert forms of probability between fractions and percentages. To address this misconception, scaffold with more familiar values initially to facilitate the interpretation of the data.

Strategies to Support Tiered Instruction

• Teacher encourages the use of precise language when working with simple experiments and repeated experiments. Students should always note when discussing “outcomes” they specify whether it is an outcome from a simple experiment, such as “heads”, or from a repeated experiment, such as “heads, heads.”
• Teacher facilitates discussion to explain that the outcome for a repeated experiment (a simple experiment that occurs more than once) consists of a sequence of outcomes that occur in the repeated simple experiment. Students should understand that the order in which the outcomes of the simple experiment occurs matters when combining to form an outcome of the repeated experiment.
  • For example, if the simple experiment is to toss a coin once, the outcomes for that simple experiment are {H, T}. If this experiment is repeated three times, there are now eight outcomes, each of which is a sequence of Hs and Ts: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Note that order matters, for instance, HHT is a different outcome than HTH or THH.
• Once students have correctly written the sample space, they can calculate the probability by counting.
  • For example, for the experiment of tossing a coin three times, the sample space can be represented as {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. If students want to determine the probability of obtaining exactly one tails in this
• experiment, they can look through the outcomes of the sample space and highlight all that have exactly one T: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
• Then, it can be observed that there are three out of eight highlighted. Therefore, the probability is $\frac{\text{3}}{\text{8}}$.
• Instruction includes the use of estimation to find the approximate decimal value of a fraction or mixed number before rewriting in decimal form to help with correct placement of the decimal point.
• Teacher co-creates a graphic organizer modeling conversions between fractions and percentages to understand and visually comprehend the relationship of the equivalent forms.
• Teacher provides opportunities for students to use a 100 frame to review place value for and the connections to decimal, fractional, and percentage forms.

Instructional Tasks

• Part A. What is the sample space?
• Part B. What is the probability the student guesses wrong answers for both questions?
• Part C. What is the probability the students guesses the correct answers for both questions?
• Part D. What is the probability the student guesses at least one correct answer.

• Part A. What is the sample space?
• Part B. Find the probability that the sum of the two results is even.

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