Final answer:
The probability that at least one out of 12 people will be infected by a cold virus, given an individual infection chance of 0.2, is approximately 0.9313 (93.13%) using the complement rule of binomial probability distribution.
Step-by-step explanation:
The probability that at least one person will be infected with the cold virus in a random sample of 12 people, when the probability of any one person being infected is 0.2, can be calculated using the complement rule. This states that the probability of at least one success is 1 minus the probability of no successes. For binomial probability distribution, where 'n' is the number of trials and 'p' is the probability of success, the probability of no one being infected (0 successes) in a sample of 12 people is given by the formula (1 - p)ⁿ.
So, the probability of no one being infected is (1 - 0.2)¹². We then subtract this value from 1 to find the probability that at least one person is infected as follows:
P(at least one infected) = 1 - (1 - 0.2)¹²
P(at least one infected) = 1 - (0.8)¹²
P(at least one infected) = 1 - 0.068719476736
P(at least one infected) = 0.931280523264
Therefore, the probability that at least one person out of a random sample of 12 people will be infected by the cold virus is approximately 0.9313 (or 93.13%).