Standard #: MA.912.DP.4.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

• Event

 

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2.2, DP.2.3, DP.2.4 
• MA.8.DP.2.2, DP.2.3

 

Next Benchmarks

 

Purpose and Instructional Strategies

In middle grades, students began working with theoretical probabilities and comparing them to experimental probability. In Mathematics for College Liberal Arts, students determine if two events are independent of each other. 

• Independence means the outcome of one event does not influence the outcome of the second event. 
  • For example, a pair of independent events are if you are rolling a die and then flipping a coin. The number on the die has no effect on whether the coin will land on heads or tails. Therefore, these events are considered to be independent. 
  • Another example would be if students draw two cards from a deck and replaced them each time from the deck, these events would be considered to be independent. 
• Dependence means that the outcome of one event influences the outcome of the second. 
  • For example, if you are drawing two cards from a deck and you do not replace them each time from the deck, these events would be considered to be dependent. 
• For this benchmark, students determine independence if the probability of Event A and B is equivalent to the probability of A times the probability of B. Students can see if events are independent by determining if P(A ∩ B) = P(A) × P(B). 
  • For example, if we roll a six-sided die and then flip a coin. These are considered to be independent events. 
    The probability of rolling a 1 is $\frac{\text{1}}{\text{6}}$. 
    The probability of a coin landing on heads is $\frac{\text{1}}{\text{2}}$. The probability of rolling a one and getting a coin landing on a head is $\frac{\text{1}}{\text{6}}$ × $\frac{\text{1}}{\text{2}}$ = $\frac{\text{1}}{\text{12}}$.
    Students can list the outcomes
    {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (4, T), (5, T), (6, T)}. 
    There are 12 outcomes equally likely to occur, therefore the probability of rolling a 1 and a landing on heads is $\frac{\text{1}}{\text{12}}$. 
    

• For example, if we were drawing socks from our sock drawer and we were not replacing them, this would be considered dependent events. Susie’s sock drawer had 3 pairs of black socks and 5 pairs of gray socks.
  
  
  
  So the probabilities of the different color combinations are:
  picking two black socks: $\frac{\text{3}}{\text{8}}$ × $\frac{\text{2}}{\text{7}}$ = $\frac{\text{6}}{\text{56}}$
  picking one black sock and one gray sock: ($\frac{\text{3}}{\text{8}}$ × $\frac{\text{5}}{\text{7}}$) + ($\frac{\text{5}}{\text{8}}$ × $\frac{\text{3}}{\text{7}}$) = $\frac{\text{30}}{\text{56}}$ 
  
  picking two gray socks: $\frac{\text{5}}{\text{8}}$ × $\frac{\text{4}}{\text{7}}$ = $\frac{\text{20}}{\text{56}}$
  Since the probability of the second pick depends on which sock you choose first, these are considered to be dependent events. 

• Be sure to distinguish independence from mutually exclusive events. Mutually exclusive events are events that cannot occur simultaneously. This is noted as P(A ∩ B) = 0. 
  • Note that P(A ∩ B) is the same as P(B ∩ A).

 

Common Misconceptions or Errors

• Students may confuse what it means to be dependent and independent. 
• Students may confuse independence with mutually exclusive events. 
• Students may struggle to convert fractions, decimals, and percentages.

 

Instructional Tasks

Instructional Task 1 (MTR.6.1) 

• One card is elected at random from a deck of 6 cards. Each card has a number and either a spade or a club: {3♣,5♣,2♠,9♣,9♠,7♣}. 
  • Part A. Let C be the event that the selected card is a club, and F be the event that the selected card is a 5. Are the events C and F independent? Justify your answer with calculation. 
  • Part B. Let S be the event that the selected card is a spade, and N be the event that the selected card is a 9. Are the events S and N independent? Justify your answer with calculation.

 

Instructional Items

Instructional Item 1 

• Probabilities for events A, B and C are described below. 
  
  P(A) = 0.20 
  P(B) = 0.55 
  P(C) = 0.36 
  P(A∩B) = 0.110 
  P(A∩C) = 0.560 
  P(B∩C) = 0.198 

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