Determine if they are independent or dependent events
A. There are 3 marbles in a bag, 2 of them are green and of them is red. You draw 2 marbles out of the bag.
B. One person in your class is chosen to lead the green team and then someone is chosen to lead the blue team.
C. P(a)=1/20 P(b)=1/2 P(a|b)=27/50
D. P(not a)=3/4 P(b)=3/10 P(a and b)=3/40
2. Assume that a and b are independent events.
A. P(a)=1/4 P(b)=3/5 P(b|a)=?
B. P(b)=9/20 P(a and b)=9/100 P(a)=?
C. P(a)=7/10 P(a or b)=173/200 P(b)=?
3. Find the missing probabilities
A. P(b)=2/5 P(and b)=1/10 P(a|b)=?
B. P(not a)=3/5 P(a and b)=6/25 P(b|a)=?
C. P(b)=0.45 P(a or b)=0.72 P(b|a)=0.4 P(a)=?
Its a. also shut up anon. No one gets pride from this mundane garbage
*psyDAG
Sure! Let's go through each question step-by-step:
1. Determine if they are independent or dependent events:
A. Drawing marbles from a bag: These events are dependent because the outcome of the first draw affects the probability of the second draw.
B. Choosing team leaders: These events are independent because the choice of the green team leader does not affect the choice of the blue team leader.
C. Using conditional probability P(a|b): For independent events, P(a|b) = P(a). However, in this case, P(a|b) is not equal to P(a), so these events are dependent.
D. Using probabilities P(a), P(b), and P(a and b): For independent events, P(a and b) = P(a) * P(b). However, in this case, P(a and b) is not equal to P(a) * P(b), so these events are dependent.
2. Assume a and b are independent events:
A. P(b|a) = P(b), since the events are independent. Therefore, P(b|a) = 3/5.
B. P(a) = P(a and b) / P(b), since the events are independent. Therefore, P(a) = (9/100) / (9/20) = 1/20.
C. P(b) = P(a or b) - P(a), since the events are independent. Therefore, P(b) = (173/200) - (7/10) = 3/40.
3. Find the missing probabilities:
A. P(a|b) = (P(a and b)) / P(b), and we have P(and b) = P(a) * P(b), which means P(a and b) = (1/10) * (2/5) = 1/25. Therefore, P(a|b) = (1/25) / (2/5) = 1/10.
B. P(b|a) = (P(a and b)) / P(a), and we have P(a and b) = (6/25), and P(not a) = 1 - P(a), so P(a) = 1 - (3/5) = 2/5. Now, P(b|a) = (6/25) / (2/5) = 3/5.
C. P(a) = P(a or b) - P(b), so we have P(a) = 0.72 - 0.45 = 0.27.
To determine whether events are independent or dependent, we need to compare the probabilities of the events occurring individually and the probability of one event occurring given that the other has already occurred.
If P(A) x P(B) = P(A and B), then the events are independent. If P(A) x P(B) does not equal P(A and B), then the events are dependent.
Let's go through each problem step by step.
1.
A. In this case, the events are dependent. To determine this, we calculate the probabilities as follows:
P(A) = 2/3 (two green marbles out of three total)
P(B) = 1/2 (one red marble out of two left after drawing a green)
P(A and B) = 2/3 x 1/2 = 1/3 (probability of drawing one green and one red)
P(A) x P(B) = 2/3 x 1/2 = 1/3
Since P(A and B) = P(A) x P(B), the events are dependent.
B. In this case, the events are independent. The choice of a leader for the green team does not affect the choice of a leader for the blue team.
C. In this case, we cannot determine whether the events are independent or dependent without more information. We can calculate:
P(A|B) = P(A and B) / P(B) = (27/50) / (1/2) = 27/25
If P(A|B) equals P(A), the events are independent. Otherwise, they are dependent.
D. In this case, the events are independent. We can calculate:
P(A) = 1 - P(not A) = 1 - 3/4 = 1/4
P(A and B) = P(A) x P(B) = (1/4) x (3/10) = 3/40
Since P(A and B) equals P(A) x P(B), the events are independent.
2.
A. Since events A and B are assumed to be independent, we can calculate P(B|A):
P(B|A) = P(A and B) / P(A) = (P(A) x P(B)) / P(A) = P(B) = 3/5
B. Using the assumption of independence, we can calculate P(A):
P(A) = P(A and B) / P(B) = (9/100) / (9/20) = 1/10
C. Using the formula P(A or B) = P(A) + P(B) - P(A and B), we can calculate P(B):
173/200 = 7/10 + P(B) - (9/100)
Solving this equation, we find P(B) = 29/100
3.
A. To find P(a|b), we need to use the formula:
P(a|b) = P(a and b) / P(b) = (P(a) x P(b|a)) / P(b)
We are missing the values for P(a) and P(b|a), so we cannot determine P(a|b) in this case.
B. Using the assumption of independence, we know that P(not a) = 1 - P(a). Thus, P(a) = 1 - P(not a) = 1 - 3/5 = 2/5.
To find P(b|a), we can use the formula:
P(b|a) = P(a and b) / P(a) = (6/25) / (2/5) = 3/10
C. With the available information, we can calculate P(a):
P(a or b) = P(a) + P(b) - P(a and b)
0.72 = P(a) + 0.45 - (0.4 x 0.45)
Solving this equation, we find P(a) = 44/100 = 11/25
Note: The missing probabilities cannot be determined without additional information or assumptions.