Of all U.S. fourth graders, 71% are assigned mathematics homework three or more times per week. If 5 fouth graders are randomly selected to be checked for this frequency of homework assignment during a randomly chosen week, find the probability that
A.) exactly one fourth grader was assigned homework three or more times that week
B.) no fourth grader was assigned homework three or more times that week
C.) at least one fourth grader was assigned homework three or more tiems that week
a)
NNNNY
NNNYN
NNYNN
NYNNN
YNNNN
so, there are five ways to sort the fourth graders
Pr(one)=5*.71*.29^4=.0251 check that.
b)Pr(none)=.29^5
c) Pr(at least one)=1-Pr(none)
A. .0251
B. .0020
C. None
To solve this problem, we can use the concept of binomial probability. Binomial probability is used to calculate the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has the same probability of success.
Given information:
- Total U.S. fourth graders: N = 100% (considering 100% as the total population)
- Percentage of fourth graders assigned homework three or more times per week: p = 71% = 0.71
Now we can proceed to calculate the probabilities using the binomial probability formula:
A) Probability of exactly one fourth grader being assigned homework three or more times:
P(X = 1) = (number of ways to choose 1 fourth grader) * (probability of success) * (probability of failure for the remaining trials)
P(X = 1) = C(5, 1) * (0.71^1) * (0.29^4)
P(X = 1) = 5 * 0.71 * 0.29^4
B) Probability of no fourth grader being assigned homework three or more times:
P(X = 0) = (probability of failure for all trials)
P(X = 0) = (0.29^5)
C) Probability of at least one fourth grader being assigned homework three or more times:
P(X >= 1) = 1 - P(X = 0)
Let's calculate the probabilities:
A) P(X = 1) = 5 * 0.71 * 0.29^4
A) P(X = 1) ≈ 0.3077
B) P(X = 0) = 0.29^5
B) P(X = 0) ≈ 0.0297
C) P(X >= 1) = 1 - P(X = 0)
C) P(X >= 1) = 1 - 0.29^5
C) P(X >= 1) ≈ 0.9703
Therefore, the probabilities are:
A) The probability that exactly one fourth grader was assigned homework three or more times that week is approximately 0.3077.
B) The probability that no fourth grader was assigned homework three or more times that week is approximately 0.0297.
C) The probability that at least one fourth grader was assigned homework three or more times that week is approximately 0.9703.
To find the probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Given:
- The percentage of U.S. fourth graders assigned math homework three or more times per week is 71%.
- 5 fourth graders are randomly selected.
- We need to calculate the probability for three scenarios:
A) Exactly one fourth grader is assigned homework three or more times.
B) No fourth grader is assigned homework three or more times.
C) At least one fourth grader is assigned homework three or more times.
Let's solve each scenario separately:
A) Exactly one fourth grader is assigned homework three or more times.
First, we need to determine the probability that a specific fourth grader is assigned homework three or more times, which is 71% or 0.71. The probability that any other fourth grader is not assigned homework three or more times is the complement, which is (1 - 0.71) = 0.29.
Since we are interested in exactly one fourth grader, we can choose any one of the five randomly selected fourth graders. Therefore, there are 5 ways to choose one fourth grader out of five.
Using the multiplication principle, we can multiply the individual probabilities for each case:
P(exactly one fourth grader assigned homework three or more times) = 5 * 0.71 * 0.29^4
B) No fourth grader is assigned homework three or more times.
Since the probability of a specific fourth grader not being assigned homework three or more times is 0.29, we can calculate the probability that none of the five fourth graders are assigned homework as:
P(no fourth grader assigned homework three or more times) = 0.29^5
C) At least one fourth grader is assigned homework three or more times.
To calculate the probability that at least one fourth grader is assigned homework three or more times, we can use the complement rule. The probability of none of the fifth graders being assigned homework is 0.29^5. Therefore, the probability that at least one fourth grader is assigned homework three or more times is:
P(at least one fourth grader assigned homework three or more times) = 1 - P(no fourth grader assigned homework three or more times) = 1 - 0.29^5
Now you can substitute the values into the formulas to find the probabilities for each scenario.