The probability that a house in an urban area will develop a leak is 5%. If 82 houses are randomly selected, what is the probability that none of the houses will develop a leak?
A) 0.050
B) 0.001
C) 0.000
D) 0.015
0.015
Correct - the answer is 0.015
Well, let's put on our clown shoes and calculate! The probability that a single house will develop a leak is 5%, which means the probability that a single house will NOT develop a leak is 95%. Since we want to find the probability that NONE of the 82 houses will develop a leak, we need to multiply the probabilities together. So it's like rolling a 95-sided dice 82 times (which is quite a workout for those tiny dice-rolling muscles)!
Mathematically, the probability of none of the houses developing a leak is (95/100)^82, which is approximately 0.001, or in other words, B) 0.001. Congrats on keeping those houses nice and dry, you leak-preventing superhero!
To find the probability that none of the houses will develop a leak, we can use the concept of independent events. Since each house is randomly selected and the probability of a house developing a leak is 5%, the probability of a house not developing a leak is 1 - 0.05 = 0.95.
Now, let's assume that each of the 82 houses is independent of each other and has the same probability of not developing a leak (0.95).
To find the probability that none of the houses will develop a leak, we need to multiply the probabilities of each house not developing a leak together. Since all the houses are independent events, we can multiply them directly:
0.95 * 0.95 * 0.95 * ... (82 times)
Written mathematically, it can be expressed as (0.95)^82.
Now, let's calculate this probability:
P(none of the houses will develop a leak) = (0.95)^82 ≈ 0.4178
So, the probability that none of the houses will develop a leak is approximately 0.4178.
Since none of the given answer choices match this probability, it seems there might be a mistake in the answer options provided.