The Better Businesss Bureau reports that they resolve 70% of the consumer complaints they receive.
a) if a random sample of 5 consumer complaints is selected, what is the probability that all 5 were resolved?
b)if a random sample of 5 consumer complaints is selected, what is the probability that at least 1 was not resolved?
c) if a random sample of 5 consumer complaints is selected, what is the probability that none were resolved?
d) if a random sample of 5 consumer complaints is selected, what is the probability that at least 1 was resolved?
Use binomial theorem with N=5, p=0.70
=> q=1-p=0.3
Let
C(n,r)=binomial coefficient=n!/(r!(n-r)!)
(a) all five resolved
P(X=5)=C(5,5)p^5q^0=1*0.7^5*1=0.17
(b) at least one not resolved
i.e. not all five
P(X≤4)=1-P(X=5)
(c) none resolved
P(X=0)=C(5,0)p^0q^5=1*1*0.3^5=0.002
(d) at least one resolved
i.e. not none resolved
P(X≥1)=1-P(X=0)
A little suggest for follow-ups to my responses.
Start with a follow-up to the original post. If your patience runs out, then start a new post with or without the recipient's name under "school subject".
Assuming you have no problem with parts (a) and (c), parts (b) and (d) are simply the complement to the preceding part.
For example,
P(X≤4) means P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4).
However, we make use of the fact that the sum of probabilities of all possible events add up to 1, or
P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)=1 (all possible events), so by transposition, we obtain:
P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)
=1-P(X=5)
as was done in (b) and (d).
To answer these probability questions, we need to use the concept of binomial probability. The binomial distribution is used when we have a fixed number of trials, each with two possible outcomes (success or failure), and the trials are independent of each other.
In this case, we have a fixed number of 5 consumer complaints, and the possible outcomes are whether each complaint is resolved or not resolved. Let's solve each question one by one:
a) To find the probability that all 5 complaints were resolved, we need to calculate the probability of success (complaint resolved) raised to the power of the number of trials (5). Given that the Better Business Bureau resolves 70% of the complaints, the probability of success is 0.7.
So, the probability that all 5 complaints were resolved is:
P(X = 5) = (0.7)^5
b) To find the probability that at least 1 complaint was not resolved, we can subtract the probability that all complaints were resolved from 1. In other words, 1 minus the probability that all 5 complaints were resolved.
So, P(at least 1 not resolved) = 1 - P(X = 5)
c) To find the probability that none of the complaints were resolved, we need to calculate the probability of failure (complaint not resolved) raised to the power of the number of trials (5). Since 70% of the complaints are resolved, the probability of failure is 30% or 0.3.
So, the probability that none of the complaints were resolved is:
P(X = 0) = (0.3)^5
d) To find the probability that at least 1 complaint was resolved, we can subtract the probability that none of the complaints were resolved from 1. In other words, 1 minus the probability that all 5 complaints were not resolved.
So, P(at least 1 resolved) = 1 - P(X = 0)
By plugging in the values and calculating the expressions, we can determine the numerical values of these probabilities.