On average, you receive four pieces of junk mail a day. For a randomsample of 10 weekends find the probability that at least 8 weekends youreceive exactly three pieces of junk mail?
To find the probability of receiving at least 8 weekends with exactly three pieces of junk mail out of a random sample of 10 weekends, we can use the binomial distribution.
The binomial distribution is used when each trial has two possible outcomes (in this case, either a weekend has exactly three pieces of junk mail or it does not), the trials are independent, and the probability of success (p) is constant.
In this case, "success" refers to a weekend with exactly three pieces of junk mail. So, we need to calculate the probability of getting 8, 9, or 10 successes.
First, let's define the variables:
n = number of trials (10 weekends)
x = number of successes (8, 9, or 10 weekends with exactly three pieces of junk mail)
p = probability of success (receiving exactly three pieces of junk mail on a weekend)
The probability of success (p) can be calculated by dividing four (average number of junk mail per day) by the total number of days in a weekend (2). So, p = 4/2 = 2.
Now, we can calculate the probability of getting exactly x successes out of n trials using the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where (nCx) is the number of combinations of n items taken x at a time, and it can be calculated as:
(nCx) = n! / (x! * (n-x)!)
So, we need to calculate P(8) + P(9) + P(10):
P(8) = (10C8) * (2^8) * (1-2)^(10-8)
P(9) = (10C9) * (2^9) * (1-2)^(10-9)
P(10) = (10C10) * (2^10) * (1-2)^(10-10)
Calculating each of these probabilities and summing them will give us the final answer.