Who is the king of late night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Suppose that you randomly select three late night TV watchers and ask them which of the two talk show hosts they prefer.
a) Find the probability distribution for x, the number of people in the sample of three who would prefer Jay Leno.
Using this formula:
P(x) = (nCx)(p^x)[q^(n-x)]
Note: q = 1 - p
We have:
P(x) = (3Cx)(.52^x)[.48^(3-x)]
To find the probability distribution for the number of people in the sample of three who would prefer Jay Leno, we can use the binomial probability formula.
Let's define "success" as a person in the sample who prefers Jay Leno. The probability of success, denoted as p, is 0.52 (based on the survey).
Now let's calculate the probability distribution for the number of people who prefer Jay Leno in the sample.
The probability of getting exactly 0 people who prefer Jay Leno is given by:
P(x = 0) = (3 choose 0) * (0.52^0) * (0.48^3)
P(x = 0) = (1) * (1) * (0.48^3) = 0.110592
Thus, the probability of getting exactly 0 people who prefer Jay Leno is 0.110592.
Similarly, we can calculate the probabilities for the other possible values of x:
P(x = 1) = (3 choose 1) * (0.52^1) * (0.48^2) = 0.406528
P(x = 2) = (3 choose 2) * (0.52^2) * (0.48^1) = 0.470016
P(x = 3) = (3 choose 3) * (0.52^3) * (0.48^0) = 0.178264
Therefore, the probability distribution for x is as follows:
x = 0, P(x = 0) = 0.110592
x = 1, P(x = 1) = 0.406528
x = 2, P(x = 2) = 0.470016
x = 3, P(x = 3) = 0.178264
To find the probability distribution for the number of people in the sample of three who prefer Jay Leno, we can use the concept of the binomial distribution.
In this case, we have a binomial distribution because we are conducting multiple independent trials (selecting three TV watchers), with each trial having two possible outcomes (preferring Jay Leno or not). The probability of success (preferring Jay Leno) is given as 0.52, based on the internet survey.
To find the probability distribution, we need to calculate the probability of each possible outcome (0, 1, 2, 3) using the formula:
P(x) = C(n,x) * p^x * (1-p)^(n-x)
Where:
- P(x) is the probability of having x individuals out of the three who prefer Jay Leno.
- C(n,x) is the number of combinations of selecting x individuals out of n.
- p is the probability of success (preferring Jay Leno).
- n is the total number of trials (in this case, three TV watchers).
Let's calculate the probabilities for each possible outcome:
P(0) = C(3,0) * 0.52^0 * (1-0.52)^(3-0) = 1 * 1 * 0.48^3 = 0.110592
P(1) = C(3,1) * 0.52^1 * (1-0.52)^(3-1) = 3 * 0.52 * 0.48^2 = 0.3456
P(2) = C(3,2) * 0.52^2 * (1-0.52)^(3-2) = 3 * 0.52^2 * 0.48^1 = 0.4488
P(3) = C(3,3) * 0.52^3 * (1-0.52)^(3-3) = 1 * 0.52^3 * 0.48^0 = 0.095808
Therefore, the probability distribution for x, the number of people in the sample of three who prefer Jay Leno, is as follows:
P(0) = 0.110592 (approximately 0.111)
P(1) = 0.3456
P(2) = 0.4488
P(3) = 0.095808 (approximately 0.096)
These values represent the probabilities of getting 0, 1, 2, or 3 people who prefer Jay Leno out of a sample of three late night TV watchers.