about 11% of the population is nervous around strangers. if 2 people are randomly selected, what is the probability that both are nervous and strangers? what is the probability that at least one is nervous around strangers?
To find the probability that both selected people are nervous around strangers, we can multiply the probability of the first person being nervous around strangers by the probability of the second person also being nervous around strangers.
Given that about 11% of the population is nervous around strangers, the probability of one person being nervous around strangers is 11% or 0.11. Since the two people are selected randomly and independently, we can use the multiplication rule of probability.
Probability that both are nervous and strangers:
P(Both are nervous) = P(First person being nervous) * P(Second person being nervous)
P(Both are nervous) = 0.11 * 0.11 = 0.0121
So, the probability that both selected people are nervous around strangers is 0.0121 or 1.21%.
To find the probability that at least one person is nervous around strangers, we can use the complement rule of probability.
Probability that at least one is nervous around strangers:
P(At least one is nervous) = 1 - P(Neither are nervous)
P(At least one is nervous) = 1 - P(Both are not nervous)
Given that about 11% of the population is nervous around strangers, the probability of one person not being nervous around strangers is 1 - 0.11 = 0.89.
So:
P(Both are not nervous) = P(First person not being nervous) * P(Second person not being nervous)
P(Both are not nervous) = 0.89 * 0.89 = 0.7921
Therefore:
P(At least one is nervous) = 1 - P(Both are not nervous)
P(At least one is nervous) = 1 - 0.7921 = 0.2079
So, the probability that at least one selected person is nervous around strangers is 0.2079 or 20.79%.
To find the probability of both events happening, we must multiply the probabilities of each event occurring separately. Let's break it down step by step.
Probability that both are nervous and strangers:
1. Calculate the probability of the first person being nervous around strangers: P(nervous and stranger) = P(nervous) * P(stranger)
- Given that 11% of the population is nervous around strangers, we can convert this percentage to a decimal: P(nervous) = 0.11
- Since we are randomly selecting people, the probability of the first person being a stranger is 100%: P(stranger) = 1.00
- Multiply these probabilities together: P(nervous and stranger) = 0.11 * 1.00
2. Calculate the probability of the second person being nervous around strangers if the first person already satisfies this condition: P(nervous and stranger) = P(nervous | first person is nervous and a stranger) * P(stranger | first person is nervous and a stranger)
- Since the first person is already nervous and a stranger, the percentage of the population becomes the new sample space.
- The probability that the second person is nervous can be assumed to be the same as the first, as the 11% figure applies to the population as a whole.
- The probability of the second person being a stranger is dependent on the presence of the first person. Assuming independence, P(stranger | first person is nervous and a stranger) remains the same as before: P(stranger | first person is nervous and a stranger) = 1.00
- Multiply these probabilities together: P(nervous and stranger) = 0.11 * 1.00
3. Multiply the probabilities of each step together to find the probability that both people are nervous around strangers:
P(both nervous and strangers) = P(nervous and stranger) * P(nervous and stranger)
- P(both nervous and strangers) = (0.11 * 1.00) * (0.11 * 1.00)
To find the probability that at least one person is nervous around strangers, we need to calculate the complement (probability of the opposite event not happening).
1. Probability that none of the two people are nervous and strangers:
- We can calculate this by subtracting the probability of at least one person being nervous and a stranger from 1, as these two events are complements of each other.
- P(none nervous and strangers) = 1 - P(both nervous and strangers)
2. Finally, to find the probability that at least one person is nervous around strangers, we subtract the probability of none being nervous and strangers from 1:
- P(at least one nervous and stranger) = 1 - P(none nervous and strangers)
Solving these equations will provide the answers to both questions.