SOLUTION: Can you answer and explain this? Please help me. Thank you! If the probabilities of A,B,C,D surviving a certain period are 3/5, 4/5, 4/5 and 7/8 respectively, what is the P that

Algebra ->  Probability-and-statistics -> SOLUTION: Can you answer and explain this? Please help me. Thank you! If the probabilities of A,B,C,D surviving a certain period are 3/5, 4/5, 4/5 and 7/8 respectively, what is the P that      Log On


   

Question 701398: Can you answer and explain this? Please help me. Thank you!
If the probabilities of A,B,C,D surviving a certain period are 3/5, 4/5, 4/5 and 7/8 respectively, what is the P that:
1. all four will survive the period
2. all four will die in this period
3. at least 1 will survive the period
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If the survival of one (or more) is somehow related to the survival of the others, we do not have enough information.
So, we have to assume that the survival of each one is independent of the survival of the others.
That means that the probability of A surviving is 3%2F5 no matter what happens to B, C, and D.
The same goes for the probability for the survival of B, C, or D.

You may have been told that events are independent events, if their joint probability is the product of the individual probabilities.
That is a hard to comprehend mouthful.

Imagine a million of parallel universes.
In a million of parallel universes, that period would happen a million times.
A would survive 3%2F5 of the million times.
B would survive 4%2F5 of the million times, and since B does not care what happens to A,
B would survive 4%2F5 of the 3%2F5 of the million times when A survived.
So survival of A and B happens %283%2F5%29%284%2F5%29=12%2F25 of the times.
The probability of A and B both surviving is the product of the probabilities for the survival of each.

1. Probability that all four will survive the period
The probability of all four surviving is the product of the probabilities for survival for each of them:
%283%2F5%29%284%2F5%29%284%2F5%29%287%2F8%29=highlight%2842%2F125%29

2. Probability that all four will die in this period
Probability that A will die is 1-3%2F5=2%2F5,
because the probability that A will survive (3%2F5),
plus the probability that A will die adds up to
the probability that A will either die, or survive, which is 1.
Similarly, the probability of dying for B, C , and D is
1%2F5, 1%2F5, and 1%2F8 respectively.
The probability of all four dying is the product of the probabilities of dying for each one:
%282%2F5%29%281%2F5%29%281%2F5%29%281%2F8%29=highlight%281%2F500%29

3. Probability that at least 1 will survive the period.
One or more of the four surviving means the opposite of all 4 dying.
Either they all 4 die, or at least one survives,
so the probability that at least 1 will survive the period is
1-1%2F500=highlight%28499%2F500%29