The probability that a house is burglarized in the city is 5%. If 50 houses are randomly selected, what is the probability that none will be burglarized?
Well, if the probability of a house being burglarized is 5%, then we can say that the probability of a house not being burglarized is 95%. So, if we randomly select 50 houses, the probability that none of them will be burglarized is (0.95)^50, which is approximately 7.62%. Now, let's hope those houses have some serious security measures... or maybe some scary clown decorations!
To find the probability that none of the 50 houses will be burglarized, we can use the probability of a single house not being burglarized.
The probability of a single house not being burglarized is given by (1 - probability of being burglarized) = (1 - 0.05) = 0.95.
Since the houses are randomly selected, the probability of none of the 50 houses being burglarized is the product of the individual probabilities.
So, the probability that none of the 50 houses will be burglarized is (0.95) raised to the power of 50.
P(none of the 50 houses are burglarized) = 0.95^50 ≈ 0.076 = 7.6% (rounded to the nearest tenth of a percent).
Therefore, the probability that none of the 50 houses will be burglarized is approximately 7.6%.
To find the probability that none of the 50 randomly selected houses will be burglarized, we can use the concept of independent events.
The probability that a single house is not burglarized is given as 1 - 0.05 = 0.95 (since the probability of a house being burglarized is 5%).
Since each house is selected independently, the probability that none of the 50 houses will be burglarized is found by multiplying the individual probabilities together.
P(none of the 50 houses are burglarized) = (0.95)^50
To calculate this probability, let me do the math for you.
Assuming a large city (so that the sample size does not affect the population probability), then the probability of a house is NOT burglarized is (1-0.05)=0.95.
By the multiplication law, randomly selecting 50 houses not burglarized is
P(50)=0.95^50=0.077.
If you work the other way round, and look for randomly selecting 50 houses ALL burglarized, the probability of that happening is then
P(50)=0.05^50=8.9*10^(-66), very small indeed.