Lesson 4.5 – The General Multiplication Rule and Tree Diagrams

  • Use the general multiplication rule to find probabilities.
  • Use a tree diagram to model a chance process involving a sequence of outcomes.
  • Calculate conditional probabilities using tree diagrams.

Activity: Can you get a pair of Aces or a pair of Kings?

Aces and KingsStart by playing a few rounds of the game in front of the whole class.  Be sure to show the students all five cards before the first draw, and all four cards before the second draw.  Ask students if getting an Ace on the first and second draw are independent events.  They should recognize that drawing cards without replacement gives us dependent events (the probability of the second card being an Ace depends on whether or not the first card is an Ace).  We also found it helpful to do one probability calculation in front of the class, such as the probability of two aces.

Students the play the game 10 times and record the humber of wins.   On the front white board, they recorded the number of wins from Aces, the number of wins from Kings, and the total number of wins.  Once all pairs had recorded the data, we found a probability of winning for the whole class.  Students then worked through the rest of the Activity in pairs.

For question #4, ask students why we are adding the probabilities.  The answer is that we can win the game by getting two Aces OR getting two Kings and those two events are mutually exclusive (Lesson 4.2).  For question #5, we are really just trying to find a conditional probability (Lesson 4.3).

Compare the answer for #4 (the calculated probability) with the experimental probability from #2.  This reminds students that probability is really just a long-run relative frequency (Lesson 4.1).  Also compare the answer for #5 to the data collected.  What percent of the wins occurred by getting Kings?

 

Notes

Help students to recognize that all the probabilities in the tree diagram for the 2nd card are actually conditional probabilities.  Also point out the fact that the sum of all the probabilities at the end of a tree diagram must add up to 1.  We require our students to calculate all of the end probabilities for any tree diagram that they create.

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