Assume that 20% of a very common insect species in your study area is parasitized. Assume that insects are parasitized independently of each other. If you collect 10 specimens of this species, what is the probability that no more than 2 specimens in your sample are parasitized?
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To calculate the probability that no more than 2 specimens in your sample are parasitized, we can use the binomial probability formula. The binomial probability formula is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability that exactly k out of n events will occur
n is the total number of trials or specimens (in this case, 10)
k is the number of successes or parasitized specimens
p is the probability of success or the probability that any given specimen is parasitized (in this case, 20% or 0.2)
(1-p) is the probability of failure or the probability that any given specimen is not parasitized (in this case, 80% or 0.8)
nCk is the number of combinations of n objects taken k at a time.
To find the probability that no more than 2 specimens in your sample are parasitized, we need to calculate the following probabilities:
P(X = 0), P(X = 1), P(X = 2).
Calculating P(X = 0):
P(X = 0) = (10C0) * (0.2^0) * (0.8^10) = 1 * 1 * 0.1073741824 = 0.1073741824
Calculating P(X = 1):
P(X = 1) = (10C1) * (0.2^1) * (0.8^9) = 10 * 0.2 * 0.134217728 = 0.268435456
Calculating P(X = 2):
P(X = 2) = (10C2) * (0.2^2) * (0.8^8) = 45 * 0.04 * 0.16777216 = 0.301989888
To find the probability that no more than 2 specimens in your sample are parasitized, you need to sum these individual probabilities:
P(no more than 2) = P(X = 0) + P(X = 1) + P(X = 2)
P(no more than 2) = 0.1073741824 + 0.268435456 + 0.301989888 = 0.6777995274
So, the probability that no more than 2 specimens in your sample are parasitized is approximately 0.6778 or 67.78%.