To determine when the Poisson distribution is a good approximation of the Binomial distribution, we should consider the conditions that traditionally make the Poisson approximation appropriate. These conditions are:
1. The number of trials [tex]\( n \)[/tex] is large.
2. The probability of success [tex]\( p \)[/tex] (or [tex]\( q \)[/tex], which is sometimes used to denote the probability of success) is small.
3. The product [tex]\( np \)[/tex] (which is the mean of the Binomial distribution) is moderate (not too large).
Given these conditions, let’s analyze each option:
1. [tex]\( n = 40 \)[/tex], [tex]\( p = 0.32 \)[/tex]:
- Here [tex]\( n \)[/tex] is moderately large, and [tex]\( p \)[/tex] is moderate but not small.
2. [tex]\( n = 40 \)[/tex], [tex]\( q = 0.79 \)[/tex]:
- Here [tex]\( n \)[/tex] is again moderately large, and [tex]\( q \)[/tex] (interpreted as the success probability [tex]\( p \)[/tex]) is quite high. This does not fit the condition of having a small probability.
3. [tex]\( n = 200 \)[/tex], [tex]\( q = 0.98 \)[/tex]:
- Here [tex]\( n \)[/tex] is very large, and [tex]\( q \)[/tex] (interpreted as the success probability [tex]\( p \)[/tex]) is close to 1, which makes the failure probability [tex]\( 1 - q = 0.02 \)[/tex] very small. This fits well into our conditions since [tex]\( p \)[/tex] is small and [tex]\( n \)[/tex] is large. Therefore, this is a good case for approximating the Binomial distribution with a Poisson distribution.
4. All of the above:
- Since not all situations (options 1 and 2) fit the traditional criteria for a good Poisson approximation, this option is incorrect.
Therefore, the best answer is (3) [tex]\( n = 200, q = 0.98 \)[/tex]. This case satisfies the conditions suitable for the Poisson approximation.
