Connecting Benchmarks/Horizontal Alignment

• MA.912.DP.3

 

Terms from the K-12 Glossary

• Event  
• Sample space 

 

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2.2
• MA.7.DP.2.3 
• MA.7.DP.2.4
• MA.8.DP.2.2
• MA.8.DP.2.3
• MA.912.DP.3.1

Next Benchmarks

 

Purpose and Instructional Strategies

In middle grades, students began working with theoretical probabilities and comparing them to experimental probability. In Algebra I students determined frequencies from two-way tables. In Math for College Liberal Arts, students calculate the conditional probability of two events and interpret the results in context of the problem.

• Conditional probability of an event is the probability that the event will occur given the knowledge that another event has already occurred. 
• Conditional probability is written as P(B|A) and is read as the probability of event B given event A. 
• Instruction includes understanding that the probability of event A given event B is different than the probability of event A given event B. Students should note that the formulas for both events are different. The conditional probability of P(B |A) = $\frac{\text{<i>P(A∩ B)</i>}}{\text{<i>P(A)</i>}}$ .
  • For example, Jackson is rolling a fair six-sided die. He wants to find the probability that the number rolled is a five, given that it is odd. The sample space for this experiment is the set S = {1, 2, 3, 4, 5, 6} consisting of six equally likely outcomes. F denotes the event “a five is rolled” and O denotes the event “an odd number is rolled.” 
    
     F = {5} and O = {1, 3, 5}
    P(F |O) = $\frac{\text{<i>P(F ∩ O)</i>}}{\text{<i>P(O)</i>}}$ 
    F ∩ O = {5} ∩ {1,3, 5}; so F ∩ O = {5}
    P(F ∩ O) = $\frac{\text{1}}{\text{6}}$ 
    Since O = {1, 3, 5}, P(O) = $\frac{\text{3}}{\text{6}}$ or $\frac{\text{1}}{\text{2}}$
    P(F |O) = $\frac{\text{<i>P(F ∩ O)</i>}}{\text{<i>P(O)</i>}}$ = $\frac{\text{1/6}}{\text{3/6}}$ = $\frac{\text{1}}{\text{3}}$
• Informally, conditional probability of P(B |A) can be thought of as restricting the sample space to only include the outcomes in event A. 
  
  • For example, Jackson is rolling a fair six-sided die. He wants to find the probability that the number rolled is a five, given that it is odd. We can reduce our sample space to the odd numbers only.
    
    O = {1, 3, 5} 
    One of these outcomes is a 5, therefore P(rolling a 5 | Odd) = $\frac{\text{1}}{\text{3}}$.
    The conditional probability of P(A|B) = $\frac{\text{<i>P(A ∩ B)</i>}}{\text{<i>P(B)</i>}}$ .
  • For example, Jackson is rolling a fair six-sided die. He wants to find the probability that the number rolled is an odd, given that it is five. The sample size for this experiment is the set S = 1, 2, 3, 4, 5, 6 consisting of six equally likely outcomes. F denotes the event “a five is rolled” and O denotes the event “an odd number is rolled.”
    
    F = 5 and O = 1, 3, 5
    P(O |F) = $\frac{\text{<i>P(O ∩ F)</i>}}{\text{<i>P(F)</i>}}$ 
    O ∩ F = 1,3,5 ∩ 5; so O ∩ F = 5
    P(O ∩ F) = $\frac{\text{1}}{\text{6}}$ 
    Since F = 5, P(F) = $\frac{\text{1}}{\text{6}}$ 
    P(O |F) = $\frac{\text{<i>P(O ∩ F)</i>}}{\text{<i>P(F)</i>}}$ = $\frac{\text{1/6}}{\text{1/6}}$ = 1
• Informally, conditional probability of P(A|B) can be thought of as restricting the sample space to only include the outcomes in event B. 
  • For example, Jackson is rolling a fair six-sided die. He wants to find the probability that the number rolled is a odd, given that it is five. We can reduce our sample space to the number 5. 
  • This one outcome is odd, therefore P(Odd | rolling a 5) = $\frac{\text{1}}{\text{1}}$ = 1. 
• Conditional probabilities can be observed in tree diagrams as the branch stemming from another branch which may be a helpful visual for some students to understand the meaning of conditional probabilities. 
• Instruction includes the use of two way tables. 
• It is not the expectation for this benchmark for students to memorize formulas.

 

Common Misconceptions or Errors

• Students may think the symbol used for conditional probability is a slash that would be used to represent division and simply divide the probability of A by the probability of B. 
• Students may get confused as to which event probability should be the denominator. 
• Students may get confused when working with a two-way table that they need to restrict their answer to a certain section that is from the “given” conditional piece. 
  • For example, when given the condition of male they are only looking in the row or column containing males to get the total. 
• Students who have difficulty with the terminology and notation will also have difficulty in understanding what is being asked by the questions.

 

Instructional Tasks

Instructional Task 1 (MTR.7.1) 

• All the students at All Florida High School were surveyed and they were classified according to year and whether or not they have one or both ears pierced. One student is randomly selected.
  
  
  

• Part A. Are the events of having your ears pierced and the year you are in school independent or dependent? How do you know? 
• Part B. What is the probability of the selected student having pierced ears? 
• Part C. What is the probability of the selected student having pierced ears given that the student is a sophomore?
• Part D. What is the probability of the selected student having pierced ears given that the student is not a sophomore? 
• Part E. What is the probability of the selected student being a sophomore given they have pierced ears?

 

Instructional Items

Instructional Item 1

• Compute the following probabilities in connection with the roll of a single fair six-sided die. 
  • Part A. The probability that the roll is even. 
  • Part B. The probability that the roll is even, given that it is not a four. 
  • Part C. The probability that the roll is even, given that it is not a three.

 

Instructional Item 2 *The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.