# Question regarding the normal approximation to the binomial probablility distribution

• Apr 26th 2010, 06:16 PM
RaVS
Question regarding the normal approximation to the binomial probablility distribution
Hello, I am really having trouble with this one:

Q.
Data collected over a long period of time show that a particular genetic defect occurs in 1 of every 1000 children. The records of a medical clinic show x=60 children with the defect in a total of 50,000 examined. If the 50,000 were a random sample from the population of children represented by past records,What is the probability of observing a value of X eqaul to 60 or more?

Any help will be greatly appreciated,
Cheers, RaVS
• Apr 27th 2010, 06:41 AM
CaptainBlack
Quote:

Originally Posted by RaVS
Hello, I am really having trouble with this one:

Q. Data collected over a long period of time show that a particular genetic defect occurs in 1 of every 1000 children. The records of a medical clinic show x=60 children with the defect in a total of 50,000 examined. If the 50,000 were a random sample from the population of children represented by past records,What is the probability of observing a value of X eqaul to 60 or more?

Any help will be greatly appreciated,
Cheers, RaVS

Assume the normal approximation is valid then the expected number with the defect is $50$ and its SD is $\sqrt{50000(1-1/1000)/1000} \approx \sqrt{50}$

CB
• Apr 27th 2010, 09:47 PM
RaVS
Thank you Captain Black, your help is greatly appreciated.
• Apr 27th 2010, 10:57 PM
matheagle
with n large and p REALLY small I would think that the Poisson limit is correct and not the normal.
• Apr 28th 2010, 12:26 AM
CaptainBlack
Quote:

Originally Posted by matheagle
with n large and p REALLY small I would think that the Poisson limit is correct and not the normal.

The condition that I use when considering the normal approximation is $\mu \gg \sigma$. In practice this means that $\sqrt{Np} \gg 1$

Depending on the time of day and wind direction this can mean anything from $\mu \ge 3 \sigma$ to $\mu \ge 10 \sigma$, or if we are discussing very low probability events $\mu \ge 100 \sigma$ to never (low probability events that occur in certain safety analyses for example).

Actually that is not what I usually do; I always first consider the normal approximation and only if the answer using that is not blindingly clear do I resort to exact calculation (or a better approximate method), that and the desire to impress simple minds. Reality does not really support high precision probability calculations, but we don't tell the punters this.

CB