Question

lesson 10-7 practice a independent and dependent events date class tell whether each set of events is independent or dependent. explain your answer. 1. you roll a number cube three times. 2. select a marble from a bag, do not replace it, then select another marble. 3. a number cube is rolled two times. a. are the events independent or dependent? b. what is the probability of rolling a 5 both times? 4. the numbers 1 – 20 are written on pieces of paper and put in a box. two pieces of paper are randomly selected and not replaced. a. are the events independent or dependent? b. what is the probability of selecting a number less than 6 both times? 5. a bag contains 1 red, 7 black, and 2 yellow marbles. state whether the following events are independent or dependent. then find the probabilities. a. probability of selecting a black marble, replacing it, then selecting a red marble b. probability of selecting a yellow marble, not replacing it, then selecting another yellow marble? c. probability of selecting 1 yellow marble, not replacing it, then selecting a black marble the number of drama club members per grade is given. two students will be chosen. drama club 9th 8 10th 2 6. what is the probability both students are 9th graders? 7. what is the probability both students are 10th graders? 8. what is the probability one student is a 9th grader and one student is a 10th grader? copyright © by hot, rinehart and winston. 51 holt algebra

Explanation:

Step1: Analyze the first event

When rolling a number cube three times, the outcome of each roll does not affect the outcome of the other rolls. The probability of getting any number on a subsequent roll remains the same as the first roll because the cube has no memory of previous rolls. So these are independent events.

Step2: Analyze the second event

Selecting a marble from a bag and not replacing it changes the composition of the bag for the next selection. The probability of selecting a particular marble on the second draw depends on what was drawn first. So these are dependent events.

Step3: Analyze the third event (a)

Rolling a number cube two times: each roll is independent of the other. The result of the first roll doesn't influence the result of the second roll. So the events are independent.

Step4: Analyze the third event (b)

The probability of rolling a 5 on a number cube is $\frac{1}{6}$. Since the rolls are independent, the probability of rolling a 5 both times is the product of the probabilities of each event. So we calculate $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

Step5: Analyze the fourth event (a)

Selecting two pieces of paper without replacement: the first selection affects the number of papers left and the composition of numbers for the second selection. So the events are dependent.

Step6: Analyze the fourth event (b)

There are 5 numbers less than 6 (1 - 5) out of 20. For the first selection, the probability is $\frac{5}{20}=\frac{1}{4}$. After one number less than 6 is selected (and not replaced), there are 4 numbers less than 6 left out of 19. So the probability is $\frac{5}{20} \times \frac{4}{19}=\frac{1}{4} \times \frac{4}{19}=\frac{1}{19}$.

Step7: Analyze the fifth event (a)

Selecting a black marble and replacing it: replacing the marble means the composition of the bag remains the same for the next selection. So the events are independent. The number of black marbles is 7, total marbles are $1 + 7+ 2 = 10$. Probability of black is $\frac{7}{10}$, probability of red is $\frac{1}{10}$. So the probability is $\frac{7}{10} \times \frac{1}{10}=\frac{7}{100}$.

Step8: Analyze the fifth event (b)

Selecting a yellow marble and not replacing it: first, number of yellow marbles is 2, total is 10. Probability of first yellow is $\frac{2}{10}=\frac{1}{5}$. After one yellow is removed, there is 1 yellow left out of 9. So probability is $\frac{2}{10} \times \frac{1}{9}=\frac{1}{5} \times \frac{1}{9}=\frac{1}{45}$. And the events are dependent.

Step9: Analyze the fifth event (c)

Selecting a yellow marble (not replacing) then a black marble: first, probability of yellow is $\frac{2}{10}=\frac{1}{5}$. After removing one yellow, total marbles are 9, black marbles are 7. Probability of black is $\frac{7}{9}$. So the probability is $\frac{2}{10} \times \frac{7}{9}=\frac{1}{5} \times \frac{7}{9}=\frac{7}{45}$. The events are dependent.

Step10: Analyze the sixth event

Total drama club members: $8 + 2=10$. Probability of first 9th grader: $\frac{8}{10}=\frac{4}{5}$. After one 9th grader is selected (not replaced), there are 7 9th graders left out of 9. So probability is $\frac{8}{10} \times \frac{7}{9}=\frac{4}{5} \times \frac{7}{9}=\frac{28}{45}$.

Step11: Analyze the seventh event

Probability of first 10th grader: $\frac{2}{10}=\frac{1}{5}$. After one 10th grader is selected (not replaced), there is 1 10th grader left out of 9. So probability is $\frac{2}{10} \times \frac{1}{9}=\frac{1}{5} \times \frac{1}{9}=\frac{1}{45}$.

Step12: Analyze the eighth event

We can calculate it in two ways: (9th then 10th) or (10th then 9th). Probability of 9th then 10th: $\frac{8}{10} \times \frac{2}{9}=\frac{4}{5} \times \frac{2}{9}=\frac{8}{45}$. Probability of 10th then 9th: $\frac{2}{10} \times \frac{8}{9}=\frac{1}{5} \times \frac{8}{9}=\frac{8}{45}$. Adding them together: $\frac{8}{45}+\frac{8}{45}=\frac{16}{45}$.

Answers:

1. Independent (because the outcome of one roll does not affect the others)
2. Dependent (because not replacing the marble changes the probability for the next draw)
3. a. Independent (each roll is independent)
b. $\frac{1}{36}$
4. a. Dependent (without replacement, the first draw affects the second)
b. $\frac{1}{19}$
5. a. Independent (with replacement, the bag remains the same); Probability: $\frac{7}{100}$
b. Dependent (without replacement); Probability: $\frac{1}{45}$
c. Dependent (without replacement); Probability: $\frac{7}{45}$
6. $\frac{28}{45}$
7. $\frac{1}{45}$
8. $\frac{16}{45}$

Answer:

s:

1. Independent (because the outcome of one roll does not affect the others)
2. Dependent (because not replacing the marble changes the probability for the next draw)
3. a. Independent (each roll is independent)

b. $\frac{1}{36}$

1. a. Dependent (without replacement, the first draw affects the second)

b. $\frac{1}{19}$

1. a. Independent (with replacement, the bag remains the same); Probability: $\frac{7}{100}$

b. Dependent (without replacement); Probability: $\frac{1}{45}$
c. Dependent (without replacement); Probability: $\frac{7}{45}$

1. $\frac{28}{45}$
2. $\frac{1}{45}$
3. $\frac{16}{45}$