Lesson 8

The mathematical purpose of this activity is to explore, formally define, and introduce notation for the concept of conditional probability.

Display the image of a deck of cards.

Explain that there are four suits (spades, hearts, diamonds, clubs) with 13 cards in each suit (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king) for a total of 52 cards. Hearts and diamonds are always considered red cards while clubs and spades are always considered black cards. Face cards are jack, queen, or king.

Unless otherwise noted, a standard deck of cards in these materials will consist of these 52 cards.

The purpose of this discussion is to formally define conditional probability and to introduce notation for conditional probability. Tell students that “conditional probability is defined as the probability of an event happening under the condition that another event occurs.” Use Elena’s situation as an example to show the notation for conditional probability: \(P(\text{the card is a queen | the card is a heart})\) which can be read as “the probability that the card is a queen under the condition that the card is a heart has occurred.” In this situation, Elena made sure the card would be a heart by removing the other suits and drawing only from those cards.

Display \(P(\text{the card is a heart | the card is a queen})\) for all to see and ask students to explain what it means, then ask students to find the probability. (It means the probability that the card is a heart under the condition that it is a queen. It has the value of \(\frac{1}{4}\) since there is one heart card among the 4 queens.) This is the notation we will use throughout this unit to represent conditional probability. Ask students “How would you write what is the probability that a card is an 8 under the condition that is a spade?” (\(P(\text{the card is an 8| the card is a spade})\)).

The mathematical purpose of this activity is to make sense of conditional probability and the relationship between conditional probabilities and the probability that two events both occur. When students do the same type of calculation several times and observe that \(P(\text{A and B}) = P(\text{A | B}) \boldcdot P(B)\) for events A and B they are engaging in MP8 because they are looking for and expressing regularity in repeated reasoning.

Remind students about the parts of a standard deck of cards including the working definition of "face card" as a jack, queen, or king of any suit. Tell students that they will be introduced to new notation for probability in the form \(P(\text{A | B})\) and explain that the vertical line inside the parentheses can be read as “under the condition that the next event occurs” or “given that the next event occurs.” For example, \(P(\text{A | B})\) can be read as "the probability of Event A under the condition that Event B occurs." Display the example for all to see.

Some students may have difficulty understanding the newly-introduced notation \(P(\text{A | B})\). Prompt them to think about how they would use the notation to write "the probability of rolling a 6 on a standard number cube under the condition that the number rolled is an even number." Emphasize that the condition of rolling an even number comes after the vertical line, and then show students the correct notation: \(P(\text{6|even number})\). In addition, prompt students to record or explain what \(P(\text{even|number is greater than 3})\) means in words. 

The purpose of this discussion is to understand how to calculate conditional probability and how conditional probability is related to independence. Ask students, “Does Kiran's equation always work? Explain your reasoning” (I think it always works because the left and right side of the equation are really two different ways of saying the same thing. If you want to know the probability of event A and event B happening then you can just look at all the probability that event A happens under the condition that event B happens then multiply by the probability of event B happening.). Confirm with students that Kiran’s equation does always work.

Ask

• “A box contains 8 crayons: 3 red, 2 blue, 1 black, 1 green, and 1 yellow. Remove one crayon from the box at random and set it apart, then remove a second crayon from the box. Use Kiran’s equation to find \(P(\text{the second crayon is blue and the first crayon is blue})\). Explain or show your reasoning.” (\(\frac{1}{28}\) since \(P(\text{the second crayon is blue | the first crayon is blue}) = \frac{1}{7}\) and \(P(\text{the first crayon is blue}) = \frac{2}{8}\).)
• “How is the probability of choosing the second crayon dependent on the first crayon?” (It is dependent because the probability that the second crayon is blue could be \(\frac{1}{7}\) or \(\frac{2}{7}\) depending on whether or not the first crayon is blue.)
• “How could you represent the crayon situation using a tree diagram?” (When you list out the probabilities of the first crayon, only 2 of the 8 are blue. For each of the picks there are 7 choices so I know there are 56 total possibilities. For each of the 2 picks that were blue 1 out 7 of the crayons that remain are blue so 2 out 56 is equal to \(\frac{1}{28}\). )

The mathematical purpose of this activity is to use conditional probability to investigate independence as it relates to the relationship of \(P(\text{A | B})\) to \(P(\text{A})\).

Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

Some students may not know what it means for events to be independent or dependent. Refer them to the previous lesson or display the definitions of independent and dependent events for all to see. The term independent events is defined as two events from the same experiment for which the probability of one event is not affected by whether the other event occurs or not, and the term dependent events is defined as two events from the same experiment for which the probability of one event depends on the result of the other event.

The purpose of this discussion is for students to understand the relationship of \(P(\text{A | B})\) to \(P(\text{A})\) for independent events.

Here are some question for discussion.

• “Think of two independent events A and B. What is \(P(\text{A | B})\) and \(P(\text{A})\)?” (I thought about event A as pulling a name out of a bag with ten different names in it and event B as drawing a face card. \(P(\text{A | B})\) is \(\frac{1}{10}\) and \(P(\text{A})\) is also \(\frac{1}{10}\).)
• “If A and B are independent events, what is the relationship of \(P(\text{A | B})\) to \(P(\text{A})\)? Explain your reasoning.” (They are equal since the probability of A happening is not influenced by whether B occurs or not.)
• “When A and B are dependent events, what is the relationship of \(P(\text{A | B})\) to \(P(\text{A})\)? Explain your reasoning” (When the event are dependent, the probabilities are not equal because event B impacts the probability of event A. If event B did not impact the probability of event A then the events would be independent.)
• Look at the values for \(P(\text{A | B})\) and \(P(\text{A | not B})\) that you found in questions 2 and 3. Why are the values equal? (These are equal because the probability of flipping heads is the same regardless of what happened with the number cube.)
• Look at the values for \(P(\text{B | A})\) and \(P(\text{B | not A})\) that you found in questions 2 and 3. Why are the values equal? (These are equal because the probability of getting a 4 is not changed by the coin showing tails.)