Standard #: MA.912.DP.4.8

• While the addition rule for probability deals with the probability that either one event or another event will occur, the multiplication rule for probability calculates the probability that two events occur at the same time. 
• Instruction includes connecting set operations (MA.912.LT.5.4) and exploring relationships using Venn Diagrams (MA.912.LT.5.5) when working to define the multiplication rule for probability: P(A and B) = P(A) · P(B |A) or P(A and B) = P(B) · P(A |B). 

• This can also be described as P(A ∩ B) = P(A) · P(B |A) = P(B) · P(A |B). 
• The probability of two events occurring at the same time leads to finding the probability one event occurs and the probability a second events occurs, given the first one already has occurred. This involves conditional probability (MA.912.DP.4.3), which leads us to the multiplication rule for probability. 
  • Rearrange the conditional probability formula below to derive the multiplication rule for probability.
    
     
  • Multiply both sides of the equation by P(B) to obtain P(A ∩ B) = P(B) · P(A |B) 
• Instruction includes a variety of real-world examples where students must determine whether the events are independent. In doing so, develop the alternate, simplified formula for finding the probability of two events occurring. 
  • If events are independent, then the probability of the first event no longer affects the probability of the second event occurring. Therefore, the formula can be simplified to P(A and B) = P(A) · P(B). 
• Tree diagrams can also be used to assist students with visualizing the multiplication rule for probability. 
  • For example, given the box below with smiley-face erasers, find the probability of choosing a yellow eraser then a blue eraser, without replacement. 

• Use the tree diagram below to determine choosing a yellow eraser and then choosing a blue eraser. 

• P(yellow and blue) = p(yellow) · P(blue | yellow)
• P(yellow and blue) = $\frac{\text{3}}{\text{10}}$ · $\frac{\text{4}}{\text{9}}$ = $\frac{\text{2}}{\text{15}}$ 
• Using a similar example, determine the probability when the events are independent. Given the box below with smiley-face erasers, find the probability of choosing 2 yellow erasers in a row, if the first eraser chosen is replaced in the box before choosing the second. 
  
  
  

• P(yellow and blue) = P(yellow) · p(yellow) 
• P(yellow and blue) = $\frac{\text{3}}{\text{10}}$ · $\frac{\text{3}}{\text{10}}$ = $\frac{\text{9}}{\text{100}}$

Common Misconceptions or Errors

• Students may not be able to determine whether two events are independent. 
• Students may incorrectly calculate conditional probability.

Instructional Tasks

• In your group take the deck of cards provided, and investigate the following outcomes: 
  • Part A. Determine how many total cards are in the deck. 
  • Part B. Determine the probability of drawing a heart out of your deck of cards. 
  • Part C. Determine the probability of drawing a 4 out of your deck of cards. 
  • Part D. Draw a 4 out of your deck, and do not put it back in to the deck. What would now be the probability of drawing a 6? Are these events independent? Explain. 
  • Part E. Determine the probability of drawing a 4 out of your deck of cards, then without replacing it, drawing a 6. 
  • Part F. What is the probability of drawing a face card, then without replacing it, drawing a second face card? Are these events dependent or independent? Justify your answer. 
  • Part G. What is the probability of drawing a face card, then replacing it, drawing a second face card? Are these events dependent or independent? Justify your answer.

Instructional Items

• If flipping a fair coin twice, what is the probability of landing on heads and then tails?

• A phone has 20 downloaded songs, 8 country and 12 rock. If the downloaded songs are played on random selection, without repeating, what is the probability the first two songs chosen will be rock?