Suppose that you are in a class of 31 students and it is assumed that approximately 13% of the population is left-handed. (Give your answers correct to three decimal places.)
(a) Compute the probability that exactly five students are left-handed.
Incorrect: 9.936
(b) Compute the probability that at most four students are left-handed.
Incorrect: .016.
(c) Compute the probability that at least six students are left-handed.
Incorrect: .197
No matter how I work these out it comes up wrong??
To calculate these probabilities, we can use the binomial probability formula. The formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (number of students in this scenario)
x is the number of successes (number of left-handed students)
p is the probability of success on a single trial (probability of being left-handed)
(1-p) is the probability of failure on a single trial (probability of not being left-handed)
(a) Compute the probability that exactly five students are left-handed:
Using the given information, we have:
n = 31
x = 5
p = 0.13 (13% expressed as a decimal)
P(5) = (31C5) * (0.13^5) * (1-0.13)^(31-5)
To calculate this manually, we must first calculate (31C5), also known as "31 choose 5":
(31C5) = 31! / 5!(31-5)! = 31! / (5! * 26!)
Now substitute the values:
P(5) = (31! / (5! * 26!)) * (0.13^5) * (0.87^26)
Using a calculator or software, you should get P(5) = 0.174.
(b) Compute the probability that at most four students are left-handed:
To find the probability at most four students are left-handed, we need to sum up the probabilities of having 0, 1, 2, 3, or 4 left-handed students.
P(at most 4) = P(0) + P(1) + P(2) + P(3) + P(4)
Similar to part (a), we will use the binomial probability formula for each value of x and calculate the probabilities. Then, sum them up to find the total probability.
(c) Compute the probability that at least six students are left-handed:
To find the probability at least six students are left-handed, we need to find the complement of the probability that less than six students are left-handed.
P(at least 6) = 1 - P(0) - P(1) - P(2) - P(3) - P(4) - P(5)
Calculate the probabilities using the binomial probability formula for each value of x and subtract them from 1 to find the complement.