A bank sells 20000 $5 notes and subsequently discovers that 200 of these notes are counterfeit. Suppose you receive 100 of these 20000 notes. Let X be the number of counterfeit notes you receive. Find P(X < 2). If you use Poisson approximation, what is the above probability approximately? Keep 4 decimal places in all your numerical results.

The probability of getting less than 2 counterfeit notes from 100 notes received can be calculated by using the base probability and applying Poisson approximation. Use the formula of Poisson distribution: P(X=k) = (e^-λ * λ^k)/k! P(X < 2) equals to the sum of P(X = 0) and P(X = 1).

To calculate the probability, we know that the bank has 200 counterfeit notes out of 20000. This gives us a base probability of 0.01 when you randomly select a note. So when you collect 100 notes the expected number (E) of counterfeit notes is 0.01 * 100 = 1. We can apply Poisson approximation as the number of trials (n) is large and the probability of success (p) in an individual trial is small.

With Poisson approximation, we need to find P(X < 2) , which is P(X = 0) + P(X = 1). We calculate these with the Poisson distribution formula P(X=k) = (e^-λ * λ^k)/k!, where λ is E, k is the number of successes. So, P(X = 0) = (e^-1 * 1^0)/0! and P(X = 1) = (e^-1 * 1^1)/1! Adding these probabilities provides the approximation of P(X < 2).

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