Are These Probability Calculation Methods Correct?

[m1359]

1. 50 vaccines are tested on patients to see how well they treat a virus. The results of the tests are recorded in this table.

___________________Positive Test Result______Negative Test Result
Vaccine is effective__________20_____________________10________
Vaccine is ineffective_________5______________________90________

(The table above doesn't really make sense to me...)

a) Show that the events the vaccine is effective and has positive result independent events. (That's how its worded and I cannot understand what it's even trying to ask... maybe you know?)

b) What is the probability that the vaccine is effective.

My attempt:
20/sample space
20/125?
= 0.16

c) What is the probability that the vaccine is effective given that the test was positive? (I don't know what this question is asking that is different from the previous question.)

d) What is the probability that the vaccine has either a positive test result or is effective?

My attempt:
P(A) = 0.16

P(B) = (11+4)/sample space
= 25/125?
= 0.2

P(A or B) = 0.16 + 0.2 = 0.36






2. Given (A ∩ B) ⊆ B ⊆ (A ∪ B), explain if the given probabilities are possible or not:

a) P(A ∪ B) = 0.6 and P(B) = 0.4
b) P(B) = 0.4 and P(A ∩ B) = 0.6
c) P(A ∪ B) = 0.4 and P(A ∩ B) = 0.6

I don't understand this question because of the (A ∩ B) ⊆ B ⊆ (A ∪ B) that I compare each question to. How do I go about solving this?

Thanks in advance.

 

[mmwave]

a) P(A ∪ B) = 0.6 and P(B) = 0.4

This is possible because P(A ∪ B) represents the probability of either event A or event B occurring, and P(B) represents the probability of event B occurring. Since event B is a subset of (or included in) event A ∪ B, it is possible for P(B) to be smaller than P(A ∪ B). In this case, there is a 40% chance of event B occurring and a 60% chance of either event A or B occurring.

b) P(B) = 0.4 and P(A ∩ B) = 0.6

This is not possible because P(A ∩ B) represents the probability of both events A and B occurring, and P(B) represents the probability of event B occurring. Since event B is a subset of (or included in) event A ∩ B, the probability of event B occurring cannot be larger than the probability of both events A and B occurring. In this case, P(B) is larger than P(A ∩ B), which is not possible.

c) P(A ∪ B) = 0.4 and P(A ∩ B) = 0.6

This is not possible because P(A ∪ B) represents the probability of either event A or event B occurring, and P(A ∩ B) represents the probability of both events A and B occurring. Since event A ∩ B is a subset of (or included in) event A ∪ B, the probability of both events A and B occurring cannot be larger than the probability of either event A or B occurring. In this case, P(A ∩ B) is larger than P(A ∪ B), which is not possible.