Independent Events
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Independent Events
Event A is said to be independent of event B is the conditional probability P(AB)is the same as the unconditional probability P(A). In other words, if the probability of event A is the same whether or not event B occurs, Event A is an independent event.
Ex. In the example from our homework of female, white smokers. Is the event of being a smoker said to be “independent” of being Caucasian? We know these aren’t causal events, but do they have some dependent relationship? There’s a mathematical way to conclude that:
P(A ∩ B)/P(B) = P(A) [B given A is the same as A itself)
= P(A ∩ B) = P(A) x P(B)
Using the stats given to us in Example 5.21 (p. 184):
P(S) = .246 (Females aged between 18 and 24 who are smokers) [For fitting into the formula above, let this be P(A)]
P (C) = .830 (Females aged between 18 and 24 who are Caucasian) [For use in the formula above, let this be P(B).]
P( S ∩ C) =.232 ( Females aged between 18 and 24 who smoke AND are Caucasian) [This is the intersection of A and B or P( A ∩ B), that which is in both Event A AND Event B.]
What is the probability of the intersection? .232
If A and B are independent events, the product of these two will NOT equal the probability of the intersection.
P(A) = .246; P(B) = .830
.246 X .830 = .204
This is NOT equal to the probability of the intersection (white AND a smoker), which was .232 Mathematically, we have shown that the event of being a smoker (A) and the event of being Caucasian (B) are NOT independent events.
What does this mean? Does it mean that being a young female smoker will always mean that the woman is white? No. It means knowing that a young female is a smoker will affect the probability of her being Caucasian. Remember the Post Hoc Fallacy from Chapter 1. Correlation does not prove causation. That’s not what you’ve demonstrated here. What you have demonstrated is that there is likely to be a strong correlation between smoking and race in young females.
Event A is independent of event B if and only if P(A|B) = P(A).
Multiplication Law of Independent Events
The probability of several independent events occurring simultaneously is the product of their separate possibilities. Ex. P(A1 ∩ A2 ∩ A3…) = P(A1) * P(A2)…P(An) if the events are independent.
The multiplication law for independent events can be applied to system reliability.
Ex. Two fully independent web servers that share no power source nor any function. Each server is said to have 99 percent reliability. What is the total system reliability?
Let F1 be the event that Server 1 fails and let F2 be the event that Server 2 fails. Then
P(F1) = 1 -.99 = .01
P(F2) = 1 -.99 = .01
Applying the rule of independence: P(F1 ∩ F2) = P(F1) * P(F2) = (.01)(.01) = .0001
The probability that at least one server is up is 1 minus the probability that both servers are down.
Or, 1 - .0001 = .9999 Dual file servers dramatically improve reliability to 99.99 percent. Whenever individual components have low reliability, overall reliability can be boosted with massiveredundancy.
Five Nines Rule
Prime business customers expect public carrier-class telecommunications data links to be available 99.999 percent of the time. It implies only 5 minutes of down time per year. High reliability of this level is also expected in mission-critical systems such as airline reservation systems or banking transfers. This is a goal to reach when planning how much redundancy may be needed to maintain this high a level of reliability.
Ex. A certain web server is up only 94 percent of the time (or down .06 of the time). How many independent servers would be required to ensure that the system is up 99.99 percent of the time? One would want to bring the probability of the servers being down to .0001. Four severs would accomplish this goal.
1 - .9999 = .0001
2 servers = (.06)(.06) = (.0036)
3 servers = (.06)(.06)(.06) = .000216
4 servers = (.06)(.06)(.06)(.06) = .00001296
The point is to bring the downtime rate to at or below .0001. Since three servers of the same performance rate (.06 downtime) brought the downtime to only .000216, a fourth was needed to reach and/or surpass the goal.
Applications of Redundancy
Professional sports teams have more players than is required given any combination at any time. Older airliners used to be suited with four engines because the older engines were less powerful and less reliable. They were able to fly even if as many as two engines went out.
Beyond the matter of individual component reliability is also the issue of cost and consequence. Cars come with only one battery because the consequence of a battery giving out is not very costly.(Walking home or having it towed). The consequence of an airliner’s engine failing is of absolutely high cost and consequence.
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