Microbes in a certain colony exposed to radiation have a 1/6 chance of being
mutated.
a. If ten microbes from the colony are sampled, what is the probability that at most 4 will be mutated?
b. If microbe from the colony is examined one by one, what is the probability that the first mutated microbe is the �fifth one that is examined?
c. If microbe from the colony is examined one by one, what is the probability that the �fifth mutated microbes is the tenth one that is examined?
a. To find the probability that at most 4 microbes will be mutated out of a sample of 10, we can calculate the sum of the probabilities of having 0, 1, 2, 3, or 4 mutated microbes.
First, let's consider the probability of having exactly k mutated microbes. The probability of a single microbe being mutated is 1/6. Therefore, the probability of a single microbe not being mutated is 1 - (1/6) = 5/6.
Using the binomial probability formula, the probability of having exactly k mutated microbes out of 10 can be calculated as:
P(k mutated) = (10 choose k) * (1/6)^k * (5/6)^(10-k)
To find the probability of at most 4 mutated microbes, we sum up the individual probabilities for k = 0, 1, 2, 3, and 4:
P(at most 4 mutated) = P(0) + P(1) + P(2) + P(3) + P(4)
You can use a calculator or software, or manually calculate the probabilities for each value of k and sum them up to find the final answer.
b. The probability that the first mutated microbe is the fifth one that is examined can be calculated using the concept of conditional probability.
The probability of selecting a mutated microbe as the first microbe is 1/6. For the first four microbes, all of them must be non-mutated, which occurs with probability (5/6)^4. Then, for the fifth microbe, it must be mutated, which occurs with probability 1/6.
Therefore, the probability can be calculated as:
P(first mutated at fifth) = (5/6)^4 * (1/6)
c. Similarly, the probability that the fifth mutated microbe is the tenth one that is examined can be calculated using conditional probability.
The probability of selecting a mutated microbe as the fifth microbe is given by (10 choose 5) * (1/6)^5 * (5/6)^5. For the previous nine microbes, all of them must be non-mutated, which occurs with probability (5/6)^9.
Therefore, the probability can be calculated as:
P(fifth mutated at tenth) = (10 choose 5) * (1/6)^5 * (5/6)^5