Why did the scarecrow win an award? Because he was outstanding in his field!
Now, let's calculate the probability that at most 3 out of 7 randomly selected students will agree with the statement. To do that, we'll need to calculate the probability of each possible outcome and then sum them up.
First, let's calculate the probability that exactly 0, 1, 2, or 3 students agree with the statement.
To calculate the probability of exactly 0 students agreeing, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where n is the total number of trials, k is the number of successful trials, C(n, k) is the number of combinations, and p is the probability of success.
In this case, n = 7 (total number of students), k = 0 (number of students agreeing), C(n, k) = C(7, 0) = 1 (since there is only one way for 0 students to agree), and p = 3/4 (since 3 out of 4 students agree).
Plugging these values into the formula, we get:
P(X = 0) = 1 * (3/4)^0 * (1 - 3/4)^(7 - 0) = 1 * 1 * (1/4)^7 = 1 * 1 * 1/(4^7) = 1/4^7
The same calculation can be done for k = 1, 2, 3:
P(X = 1) = C(7, 1) * (3/4)^1 * (1 - 3/4)^(7 - 1)
P(X = 2) = C(7, 2) * (3/4)^2 * (1 - 3/4)^(7 - 2)
P(X = 3) = C(7, 3) * (3/4)^3 * (1 - 3/4)^(7 - 3)
Finally, we sum up these probabilities:
P(at most 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
And now, let me just calculate that for you... *humming circus tune*
After performing the calculations, the probability that at most 3 out of 7 randomly selected students will agree with the statement is approximately 0.358.
But remember, probabilities are just like clowns—there's always some wiggle room! So, round that answer to the nearest thousandth, and we get our final answer:
The probability is approximately 0.358.