a. We can use the Poisson distribution to model the probability of counting a certain number of RBCs in a grid square, given the average number (λ) is 2.1. The probability mass function (PMF) for the Poisson distribution is:
P(X = k) = (e^(-λ) * (λ^k)) / k!
where X is the number of RBCs, k is the specific number of RBCs we are interested in, λ is the average number of RBCs (2.1 in this case), and e is the base of the natural logarithm (approximately 2.71828).
For this part, we want to find the probability that there will be no RBCs (k=0) counted in a grid square, so:
P(X = 0) = (e^(-2.1) * (2.1^0)) / 0! = 0.1225
So, the probability of counting no RBCs in a grid square is approximately 12.25%.
b. To find the probability of counting at most 3 RBCs in a grid square, we add the probabilities of counting 0, 1, 2, or 3 RBCs:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Using the Poisson PMF:
P(X = 1) = (e^(-2.1) * (2.1^1)) / 1! = 0.2573
P(X = 2) = (e^(-2.1) * (2.1^2)) / 2! = 0.2699
P(X = 3) = (e^(-2.1) * (2.1^3)) / 3! = 0.1887
P(X ≤ 3) = 0.1225 + 0.2573 + 0.2699 + 0.1887 = 0.8384
So, the probability of counting at most 3 RBCs in a grid square is approximately 83.84%.
On an average, the number of red blood cells per grid square in a hemocytometer is 2.1.
a. What is the probability that there will be no RBCs counted in a grid square?
b. What is the probability of counting at most 3 RBCs in a grid square?
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