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Common Core Standard HSS-CP.A.3 Questions

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

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Grade 10 Represent and Determine Probability CCSS: HSS-CP.A.3
There are two dependent events, A and B. The probability of event A occurring, given that event B occurs, is 0.66. The probability that events A and B occur is 0.55. Find P(A) and P(B) to two decimal places.
  1. P(A) = 0.83, P(B) = 0.66
  2. P(A) = 0.66, P(B) = 0.83
  3. P(A) = 0.66, P(B) = unknown
  4. P(A) = unknown, P(B) = 0.83
Grade 10 Represent and Determine Probability CCSS: HSS-CP.A.3
Adam has a bag of ten marbles, of which 3 are green, 3 are blue, and 4 are red. He also has a regular six-sided die. He lets choosing a blue marble be event A, and throwing a 5 or 6 when rolling the die be event B. Which of the following statements are correct? There may be more than one correct answer.
  1. P(A) = P(A|B), since events A and B are independent.
  2. P(A and B) = 0.1, since A and B are independent.
  3. P(A|B) = P(B|A), since events are independent.
  4. P(A|B), P(B|A) cannot be determined since not enough information has been given to determine P(A and B).
Grade 10 Represent and Determine Probability CCSS: HSS-CP.A.3
Mary is doing an experiment where she chooses two marbles, one after the other without replacement, from a bag of marbles. The bag of consists of 5 green, 6 red, and 9 blue marbles. She wants to find the probability that she chooses a blue marble given that she chose a green marble first. She lets G be the event that she chooses a green marble and B be the event that she chooses a blue marble. Mary reasons that, since [math]P(G) = 1/4[/math] and [math]P(B) = 9/20[/math], [math]P(G " then " B) = 9/80[/math]. (She uses "then" instead of "and" because the events happen one after the other). Therefore, she determines that [math]P(B|G) = 9/20[/math]. Is she correct, and why?
  1. Yes, her assumptions and steps are all correct.
  2. No, she assumed that G and B are independent when they are not. [math]P(G " then " B) = 9/76[/math], and therefore [math]P(B|G) = 9/19[/math].
  3. No, she calculated P(G then B) incorrectly. [math]P(G " then " B) = 14/20 = 7/10[/math], and therefore [math]P(B|G) = 14/5[/math].
  4. No, she found [math]P(G " then " B)[/math], when this is unnecessary. Since the events are independent, she can simply say [math]P(G) = P(G|B)[/math].
Grade 10 Represent and Determine Probability CCSS: HSS-CP.A.3
Aaron is given the following situation. There are 35 students in a class, 7 of which have a 90+ grade average. He chooses two students at random, one after the other. Let choosing a student with a 90+ grade average be event A, and choosing a student with an average below a 90 be event L. He wants to know whether these events are independent or dependent. Which is it, and why? Choose all correct answers.
  1. The events are dependent. After the first student has been chosen, there are fewer students, and so the probability of choosing the second student will be different than if the second student had been chosen from the full group of students.
  2. The events are dependent. Since [math]P(L) = 4/5[/math], [math] P(L|A) = 14/17[/math], and these are not equal, the events are dependent.
  3. The events are dependent. [math]P(A " then " L) = 3/85[/math] and [math]P(A) * P(L) = 4/25[/math]. Since these are not equal, the events are not independent, and must be dependent.
  4. The events are independent. Because choosing the first student as a 90+ average student doesn't change the number of students with an average below 90, the probability will remain the same regardless of the first event.

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