HI, I don't know how to approach this problem, could someone please help, thank you.
Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large-scale experiment to determine what dilution of the fungicide to apply, six treated seeds and five untreated seeds were planted in clay soil and the number of plants emerging from the treated and untreated seeds were recorded. Suppose the dilution was not effective and only four plants emerged. Let x represent the number of plants that emerged from treated seeds. (Round your answers to three decimal places.)
(a) Find the probability that x = 4.
(b) Find
P(x ¡Ü 3).
(c) Find
P(2 ¡Ü x ¡Ü 3).
I need help to solve the question
To solve this problem, we need to use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
where:
- P(x) is the probability of x successes
- n is the number of trials
- x is the number of successes
- p is the probability of success in a single trial
(a) Find the probability that x = 4:
In this case, n is the total number of seeds planted, which is the sum of treated and untreated seeds: n = 6 + 5 = 11.
Since the probability of success is the same for each trial (the emergence of a plant), we can use the same probability, which is the proportion of treated seeds that produced plants: p = 4/11 = 0.3636.
Using the binomial probability formula with x = 4, we can calculate the probability:
P(x = 4) = (11C4) * (0.3636)^4 * (1 - 0.3636)^(11 - 4)
Calculating this expression will give you the answer.
(b) Find P(x ≤ 3):
To find this probability, we need to calculate the probabilities for x = 0, 1, 2, and 3, and then add them up.
P(x ≤ 3) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)
Using the binomial probability formula, calculate each of these probabilities and sum them up to find P(x ≤ 3).
(c) Find P(2 ≤ x ≤ 3):
To find this probability, we need to calculate the probabilities for x = 2 and x = 3, and then add them up.
P(2 ≤ x ≤ 3) = P(x = 2) + P(x = 3)
Using the binomial probability formula, calculate each of these probabilities and add them up to find P(2 ≤ x ≤ 3).
To solve this problem, we can use the concept of probability and the binomial distribution. The binomial distribution is used when we have a fixed number of trials, in this case, planting seeds, and each trial can result in one of two outcomes, in this case, either a plant emerging or not.
(a) To find the probability that x = 4, we need to determine the probability of exactly 4 plants emerging from the treated seeds.
In this scenario, we planted 6 treated seeds, and we know that only 4 plants emerged.
Using the binomial probability formula, we have:
P(x = 4) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- n is the number of trials (6, in this case)
- x is the number of successful outcomes (4, in this case)
- C(n, x) is the combination formula which represents the number of ways to choose x successes from n trials
- p is the probability of success in a single trial (unknown in this case)
Since we are given that only 4 plants emerged, we can set up the following equation:
P(x = 4) = C(6, 4) * p^4 * (1-p)^(6-4)
Simplifying this equation:
P(x = 4) = 15 * p^4 * (1-p)^2
(b) To find P(x ≤ 3), we need to calculate the probability of having 3 or fewer plants emerging from the treated seeds.
P(x ≤ 3) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)
Using the same binomial probability formula, we can calculate the probabilities for each individual case and then add them together.
P(x ≤ 3) = [C(6, 0) * p^0 * (1-p)^(6-0)] + [C(6, 1) * p^1 * (1-p)^(6-1)] + [C(6, 2) * p^2 * (1-p)^(6-2)] + [C(6, 3) * p^3 * (1-p)^(6-3)]
(c) To find P(2 ≤ x ≤ 3), we need to calculate the probability of having 2 or 3 plants emerging from the treated seeds.
P(2 ≤ x ≤ 3) = P(x = 2) + P(x = 3)
Using the same binomial probability formula as before, we can calculate the probabilities for each individual case and then add them together.
P(2 ≤ x ≤ 3) = [C(6, 2) * p^2 * (1-p)^(6-2)] + [C(6, 3) * p^3 * (1-p)^(6-3)]
To find the exact values of probabilities (a), (b), and (c), we need to know the value of p, which represents the probability of success in a single trial, i.e., the probability of a seed germinating. This information is not provided in the question, so we cannot determine the exact probabilities without this information.