Suppose that you are in a class of 29 students and it is assumed that approximately 16% 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.
(b) Compute the probability that at most four students are left-handed.
(c) Compute the probability that at least six students are left-handed.
Formula:
P(x) = (nCx)(p^x)[q^(n-x)]
For (a):
x = 5
p = .16
q = 1 - p = 1 - .16 = .84
n = 29
Substitute and calculate for your probability.
For (b):
x = 0,1,2,3,4
p,q,n stay the same.
Add each probability you calculate for the total probability.
For (c):
Add together the probabilities you calculated for (a) and (b). Then subtract that value from 1. This will be your answer for (c).
You could also use a binomial probability table to find the above probabilities as well. This would be easier than calculating each by hand.
I hope this will help get you started.
To solve these probability problems, we can use the binomial probability formula. The formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k "successes" (in this case, left-handed students).
- C(n,k) is the number of combinations of n items taken k at a time, also known as the binomial coefficient.
- p is the probability of a single success (in this case, the probability of being left-handed).
- n is the total number of trials (in this case, the number of students).
- k is the number of desired successes.
Let's proceed to calculate the values for each question:
(a) Compute the probability that exactly five students are left-handed.
Using the given information:
- p = 0.16 (probability of being left-handed)
- n = 29 (total number of students)
- k = 5 (number of desired left-handed students)
Using the binomial probability formula, we can calculate as follows:
P(X=5) = C(29, 5) * 0.16^5 * (1-0.16)^(29-5)
(b) Compute the probability that at most four students are left-handed.
To calculate this, we need to find the sum of the probabilities of having 0, 1, 2, 3, or 4 left-handed students.
P(X<=4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)
We can calculate each probability using the binomial formula mentioned earlier.
(c) Compute the probability that at least six students are left-handed.
To calculate this, we need to find the sum of the probabilities of having 6, 7, 8, ..., up to 29 left-handed students.
P(X>=6) = P(X=6) + P(X=7) + ... + P(X=29)
Again, we can calculate each probability using the binomial formula.