A survey shows that 75% of the households in a large town have microwave ovens. If 20 houses are selected at random, find the following: just set up. Do NOT evaluate. A) the probability that no more than one of these households has a microwave oven. B) the probability that at least 3 of these households have a microwave oven. For part a, is it probability of both x=0 and x=1 subtracted from one? I always get confused when I see the phrase "no more than". And for part b is it just exactly p(x=3) since it has the words at least?
Yes, you are on the right track for both parts of the question.
For part A, when it says "no more than one," it means you need to calculate the probability of the event occurring when there are either zero or one households with a microwave oven. So, you are correct, you would find the probability of both x=0 and x=1 and subtract it from one.
For part B, when it says "at least three," it means you need to calculate the probability of the event occurring when there are three or more households with a microwave oven. So, in this case, you would find the probability of exactly x=3, plus the probability of x=4, x=5, and so on, until the highest possible value.
In both cases, you should set up the probability calculation using the binomial distribution formula. The formula is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of x households having a microwave oven
- n is the total number of households (20 in this case)
- x is the number of households having a microwave oven
- p is the probability of a household having a microwave oven (75% or 0.75 in this case)
- C(n, x) is the combination function that calculates the number of ways x items can be chosen from a set of n items.
After setting up the formula with the given values, you can simplify it and leave it in terms of x without evaluating the expression.