Question

On May 13, 2013, starting at 4:30 PM, the probability of moderate seismic activity for the next 48 hours in the Kuril Islands off the coast of Japan was reported at about 1.43%. Use this information for the next 100 days to find the probability that there will be low seismic activity in five of the next 100 days. Use both the binomial and Poisson

distributions to calculate the probabilities. Are they close?

Short answer

$P(x=5)=\text{binompdf}(100,0.0143,5)\approx 0.01146$

$P(x=5)=\text{poissonpdf}(1.43,5)\approx 0.01192$

Step-by-step solution

Given information

the probability of moderate seismic activity for the next 48 hours in the Kuril Islands off the coast of Japan was reported at about 1.43%.

Use this information for the next 100 days to find the probability that there will be low seismic activity in five of the next 100 days.

Explanation

Let X = the number of days with low seismic activity.

Using the binomial distribution:

$P(x=5)=\text{binompdf}(100,0.0143,5)\approx 0.01146$

Using the Poisson distribution: $\mu =np=100\left(0.0143\right)\approx 1.43$

$P(x=5)=\text{poissonpdf}(1.43,5)\approx 0.01192$

We expect the approximation to be good because n is large (greater than 20) and p is small (less than 0.05). The results are close—both probabilities reported are almost 0.