In a random poll of science students, it was determined that 90% of students submitted tests before the deadline.
In a random sample of 75, what is the probability that exactly 10 submitted after the deadline? What is the probability that less than 10 submitted after the deadline?
To solve these probability questions, we can use the binomial probability formula. In this case, the formula can be used to determine the probability of a certain number of successes (students submitting tests after the deadline) in a given number of trials (random sample of science students).
The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time (also known as "n choose k"),
p is the probability of success (submitting after the deadline), and
n is the total number of trials (sample size).
Let's use this formula to calculate the probabilities:
1. Probability of exactly 10 students submitting after the deadline:
To solve this, we can substitute the given values into the formula:
n = 75 (sample size),
k = 10 (number of successes),
p = 0.1 (probability of success).
Thus, P(X = 10) = C(75, 10) * 0.1^10 * (1 - 0.1)^(75 - 10).
2. Probability of less than 10 students submitting after the deadline:
To find the probability of less than 10 students submitting after the deadline, we need to calculate the sum of probabilities from 0 to 9. We can do this by evaluating P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 9).
Now, to calculate the probabilities, we need to use combinations. The formula for combinations, C(n, k), is:
C(n, k) = n! / (k!(n - k)!)
Where "!" denotes factorial. For example, 75! = 75 * 74 * 73 * ... * 3 * 2 * 1.
By substituting the appropriate values into the formulas, we can calculate the probabilities.