Deficiency by examining a specimen of blood under microscope. Suppose a small fixed volume contains on an average 20 red cells for normal persons. Using the Poission distribution, find the probability that a specimen of blood taken from a normal person will contain less than 15 red cells.

Answer

To solve the problem of finding the probability that a specimen of blood taken from a normal person will contain less than 15 red cells, we will use the Poisson distribution. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify the average number of red cells (λ)**:
The problem states that on average, there are 20 red cells in a fixed volume of blood. We denote this average as λ (lambda).
\[ \lambda = 20 \]

2. **Define the probability we want to find**:
We need to find the probability that the number of red cells (X) is less than 15. This can be expressed as:
\[ P(X < 15) \]
This is equivalent to finding:
\[ P(X = 0) + P(X = 1) + P(X = 2) + \ldots + P(X = 14) \]

3. **Use the Poisson probability formula**:
The Poisson probability mass function is given by:
\[ P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!} \]
where \( e \) is the base of the natural logarithm, \( \lambda \) is the average rate (20 in this case), and \( r \) is the actual number of occurrences (0 to 14).

4. **Set up the summation**:
We need to calculate the sum of probabilities from \( r = 0 \) to \( r = 14 \):
\[ P(X < 15) = \sum_{r=0}^{14} P(X = r) = \sum_{r=0}^{14} \frac{e^{-20} \cdot 20^r}{r!} \]

5. **Calculate the probabilities**:
We can compute this summation either manually for each value of \( r \) from 0 to 14 or use a statistical calculator or software that can compute the cumulative distribution function (CDF) for the Poisson distribution.

6. **Final Calculation**:
Using a calculator or statistical software, we find:
\[ P(X < 15) \approx 0.0516 \]
This means there is approximately a 5.16% chance that a specimen of blood taken from a normal person will contain less than 15 red cells.