Final answer:
The questions ask about computing probabilities for various situations, which require understanding the probability of events and the use of random variables. Examples provided illustrate the use of probability and random variable concepts in real-world scenarios.
Step-by-step explanation:
The questions provided relate to the concept of probability in mathematics. Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1. Here are some examples to illustrate these concepts using the given information:
- To determine the probability that all three used drugs, we would need data indicating the frequency of drug use among the three subjects. Without this data, we cannot calculate the required probability.
- The probability that at least one used drugs includes any scenario where one, two, or all three subjects used drugs. Assuming independence, this can be calculated using the complement of the probability that none used drugs (1 - P(no one used drugs)).
- For exactly one subject using drugs, you would again need the probability of drug use for each individual, and the calculation would involve summing the probabilities of each scenario where only one subject used drugs.
In the case of random variables, X represents the number of successes in a given scenario. For example, in the given community, 65 percent of households include at least one college graduate. The mean number of such households in a sample of 100 would be 65, as X represents the number of households with at least one college graduate, and the probability distribution of X would be binomial. The standard deviation would then be calculated based on the binomial distribution formula.