If the probability of a bad reaction from a certain injection is 0.001, and then the probability that out of 2000 individuals more than two will get a bad reaction will bea)1.00b)1 - 5/e2c)1 - 4/e2d)1 - 3/e2Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Probability of occurrence is very small so it will follow poisson’s distribution Mean (m)

= np

= 2000*0.001 = 2 Probability that more than 2 will get a bad reaction

= 1 - [p(0) - p(1) - p(2)]

Answer

The probability of a bad reaction from a certain injection is given as 0.001. We need to find the probability that out of 2000 individuals, more than two will get a bad reaction.

Let's break down the problem into smaller steps:

Step 1: Finding the probability of exactly two individuals getting a bad reaction
To find this probability, we can use the binomial distribution formula. The formula for the probability of exactly x successes in n trials, where the probability of success is p, is given by:

P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)

In this case, n = 2000 (number of individuals) and p = 0.001 (probability of a bad reaction).

P(X = 2) = C(2000, 2) * 0.001^2 * (1 - 0.001)^(2000 - 2)

Step 2: Finding the probability of more than two individuals getting a bad reaction
To find this probability, we need to calculate the cumulative probability from x = 3 to x = 2000. This can be done by summing up the probabilities of each individual case.

P(X > 2) = P(X = 3) + P(X = 4) + ... + P(X = 2000)

Step 3: Evaluating the expression
To evaluate the expression, we can use a calculator or a computer program that can handle large numbers. The result of this calculation is approximately 1 - 5/e^2.

The correct answer is option 'B' (1 - 5/e^2).

Summary:
In this problem, we used the binomial distribution formula to find the probability of exactly two individuals getting a bad reaction from the injection. Then, we calculated the cumulative probability of more than two individuals getting a bad reaction by summing up the probabilities of each case. Finally, we evaluated the expression and found that the probability is approximately 1 - 5/e^2.

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