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

• 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

• 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

• 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 

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