# Conditional Probability (Grade 10)

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## Conditional Probability

1.

If P(A) = 0.23, P(B) = 0.64, and P(A|B) = 0.23, then events A and B are independent.

- True
- False

2.

If P(A) = 0.36, P(B) = 0.44, and P(A|B) = 0.44, then events A and B are independent.

- True
- False

3.

Mark is doing an experiment. He has called two of the events H and W. Mark determines that the probability of event H occurring is 38%, the probability of event W occurring is 91%, and the probability that event W occurs, given that event H has occurred, is 35%. Therefore, since [math]0.38*0.91~~0.35[/math] he concludes that the two events are independent. Is this true or false?

- True
- False

4.

For a given experiment, two of the events are called A and B. If P(A) = 0.86, P(B) = 0.18, and P(A and B) = 0.11, what is P(A|B) about equal to?

- 0.86
- 0.18
- 0.13
- 0.61

5.

In an experiment, the probability of event A is 0.83, the probability of event B is 0.12, and the probability of event B given event A occurs is 0.55. What is the probability that event A and B occur, to two decimal places?

- 0.10
- 0.55
- 0.46
- 0.95

6.

At Withrow High School, 36% of male students participate in volleyball. There are two parts of the volleyball team, a varsity squad and junior varsity squad. Cameron knows that, if he makes the team, he has about a 30% chance of joining the varsity squad. What is the probability, to the nearest whole number, of Cameron playing volleyball and making the varsity squad?

- 11%
- 30%
- 70%
- 25%

7.

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.

- P(A) = 0.83, P(B) = 0.66
- P(A) = 0.66, P(B) = 0.83
- P(A) = 0.66, P(B) = unknown
- P(A) = unknown, P(B) = 0.83

8.

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.

- P(A) = P(A|B), since events A and B are independent.
- P(A and B) = 0.1, since A and B are independent.
- P(A|B) = P(B|A), since events are independent.
- P(A|B), P(B|A) cannot be determined since not enough information has been given to determine P(A and B).

9.

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?

- Yes, her assumptions and steps are all correct.
- 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].
- 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].
- 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].

10.

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.

- 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.
- 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.
- 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.
- 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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