Need help please on these questions:
1. A poll at Steve's high school was taken to see if students are in favor of spending class money to expand the junior-senior parking lot. Steve surveyed 6 random students from the population. 85% favored the expansion, while 15% opposed it.
a) Determine the probabilities associated with the number of students that Steve asked who are in favor of expanding the parking lot by calculating the probability distribution.
b) What is the probability that no more than 2 people are in favor of expanding the parking lot?
c) How many students should Steve expect to find who are in favor of expanding the parking lot?
2. You have a bag of marbles containing 4 blue marbles, 5 red marbles, and 3 green marbles. What is the theoretical probability of pulling a red marble? Run an experiment to see how long it takes you to pull a red marble (you don't have to use marbles) and see how the experimental and theoretical compare.
1. To determine the probabilities associated with the number of students that Steve asked who are in favor of expanding the parking lot, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
In this case, n represents the number of students Steve surveyed, k represents the number of students in favor of expanding the parking lot, and p is the probability of a student being in favor.
a) To calculate the probability distribution, we need to calculate the probabilities for each possible value of k (0, 1, 2, 3, 4, 5, and 6) using the given values. The probability of a student being in favor is 85% or 0.85.
For k = 0:
P(X = 0) = C(6, 0) * 0.85^0 * (1 - 0.85)^(6 - 0) = 1 * 1 * 0.15^6 = 0.0156
Similarly, calculate the probabilities for k = 1, 2, 3, 4, 5, and 6 using the same formula.
b) To find the probability that no more than 2 people are in favor of expanding the parking lot, we need to sum the probabilities for k = 0, 1, and 2:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Simply add these probabilities together.
c) To determine the expected number of students Steve should find who are in favor of expanding the parking lot, we can use the expected value formula for a binomial distribution:
E(X) = n * p
In this case, n represents the number of students Steve surveyed (6) and p is the probability of a student being in favor (0.85).
Simply multiply these values together to find the expected number.
2. The theoretical probability of pulling a red marble can be calculated by dividing the number of desired outcomes (red marbles) by the total number of possible outcomes (all marbles).
The number of red marbles is 5, and the total number of marbles is 4 (blue) + 5 (red) + 3 (green) = 12.
The theoretical probability of pulling a red marble is:
P(red) = 5 / 12 = 0.4167 (rounded to four decimal places)
To run an experiment and compare the experimental and theoretical probabilities, you can follow these steps:
1. Gather a set of marbles that matches the number and colors mentioned (4 blue, 5 red, 3 green).
2. Randomly select one marble from the bag without looking.
3. Record the color of the marble and repeat steps 2 and 3 until a red marble is selected.
4. Repeat the experiment multiple times (e.g., 100 times).
5. Calculate the experimental probability of pulling a red marble by dividing the number of successful outcomes (red marbles) by the total number of trials.
6. Compare the experimental probability with the theoretical probability calculated earlier.
By repeating the experiment, you can observe how the experimental and theoretical probabilities compare and determine whether they are close or deviate significantly.