37% of college students say they use credit cards because of the rewards program. You randomly select 10 college kids and ask each to name the reason he or she uses credit cards. Find the probability that the number of college students who say they use credit cards because of the rewards program is (a) exactly two, (b) more than two, and (c) between two and five inclusive.
44% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite nut is (a) exactly three, (b) at least four, and (c) at most two.
To find the probability for different scenarios, we can use the binomial probability formula.
The probability for a specific outcome with x successes in n trials is given by the formula:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- nCx is the number of combinations of n items taken x at a time
- p is the probability of success
- q is the probability of failure (1 - p)
Given:
- n = 10 (the number of college students randomly selected)
- p = 0.37 (the probability of a college student using credit cards because of the rewards program)
- q = 1 - p = 1 - 0.37 = 0.63
(a) To find the probability that exactly two college students say they use credit cards because of the rewards program, we use the binomial probability formula for x = 2:
P(2) = (10C2) * (0.37^2) * (0.63^(10-2))
(b) To find the probability that more than two college students say they use credit cards because of the rewards program, we need to calculate the probabilities for x = 3, 4, 5, 6, 7, 8, 9, and 10. Then, we sum up these individual probabilities:
P(more than 2) = P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
(c) To find the probability that between two and five college students (inclusive) say they use credit cards because of the rewards program, we need to calculate the probabilities for x = 2, 3, 4, and 5. Then, we sum up these individual probabilities:
P(between 2 and 5 inclusive) = P(2) + P(3) + P(4) + P(5)
Now, let's calculate these probabilities.
To find the probability of the number of college students who say they use credit cards because of the rewards program, we can use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of exactly k successes,
- n is the total number of trials (sample size),
- k is the number of successes,
- p is the probability of success on a single trial (37% or 0.37 in this case),
- (nCk) is the binomial coefficient, or the number of ways to choose k successes from n trials,
- (1 - p) is the probability of failure on a single trial.
(a) To find the probability that exactly two college students say they use credit cards because of the rewards program, we substitute the values into the formula:
P(X = 2) = (10C2) * (0.37)^2 * (1 - 0.37)^(10 - 2)
= (10! / (2! * (10 - 2)!)) * (0.37)^2 * (0.63)^8
= (10 * 9 / (2 * 1)) * (0.37)^2 * (0.63)^8
= 45 * (0.37)^2 * (0.63)^8
= 0.2150 (rounded to four decimal places)
Therefore, the probability that exactly two college students say they use credit cards because of the rewards program is 0.2150 or 21.50%.
(b) To find the probability that more than two college students say they use credit cards because of the rewards program, we need to calculate the probabilities for three, four, five, six, seven, eight, nine, and ten students.
P(X > 2) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
= [(10C3) * (0.37)^3 * (0.63)^7] + [(10C4) * (0.37)^4 * (0.63)^6] + [(10C5) * (0.37)^5 * (0.63)^5]
+ [(10C6) * (0.37)^6 * (0.63)^4] + [(10C7) * (0.37)^7 * (0.63)^3] + [(10C8) * (0.37)^8 * (0.63)^2]
+ [(10C9) * (0.37)^9 * (0.63)^1] + [(10C10) * (0.37)^10 * (0.63)^0]
Calculating this sum will give you the probability that more than two college students say they use credit cards because of the rewards program.
(c) To find the probability that the number of college students who say they use credit cards because of the rewards program is between two and five (inclusive), we need to calculate the probabilities for two, three, four, and five students.
P(2 ≤ X ≤ 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
This will give you the probability that the number of college students falls within the desired range.
those who use it -- .37
those who don't --- .63
prob(exactly 2 out of 10 will use it)
= C(10,2) (.37)^2 (.63)^8 = .003397
b)
more than two ---> exclude cases of none, one, and two
prob = 1 - ( C(10,0)( .37)^0 (.63)10 + C(10,1) (.37)(.63)^9 + C(10,2) (.37)^2 (63)^8 )
= ...
you do the button pushing
c) repeate abvove method for
two cases + three cases + four cases + five cases