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In a certain college class, 40 percent of students scored an A on the final exam. What is the probability of randomly selecting two out of six students from the class who scored an A? Use the formula nCkpkqn−k

to solve.(1 point)
Responses

2 percent
2 percent

4 percent
4 percent

31 percent
31 percent

14 percent

3 answers

To solve this problem, we need to find the probability of randomly selecting exactly 2 students who scored an A from a group of 6 students, where 40% of students scored an A.

Let's denote:

  • \( n = 6 \): the total number of students
  • \( k = 2 \): the number of students who scored an A
  • \( p = 0.4 \): the probability of a student scoring an A
  • \( q = 1 - p = 0.6 \): the probability of a student not scoring an A

We can use the formula for combinations and the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

First, we need to calculate the binomial coefficient \( \binom{6}{2} \):

\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

Next, we apply the probabilities:

\[ p^k = (0.4)^2 = 0.16 \] \[ q^{n-k} = (0.6)^{6-2} = (0.6)^4 = 0.1296 \]

Now we can plug these values into the probability formula:

\[ P(X = 2) = \binom{6}{2} p^2 q^{4} = 15 \times 0.16 \times 0.1296 \]

Calculating this step-by-step:

  1. Calculate \( 15 \times 0.16 \): \[ 15 \times 0.16 = 2.4 \]

  2. Calculate \( 2.4 \times 0.1296 \): \[ 2.4 \times 0.1296 \approx 0.310656 \]

Finally, express this in percentage:

\[ 0.310656 \times 100 \approx 31.07% \]

Thus, the probability of randomly selecting exactly 2 students who scored an A is approximately 31%.

Therefore, the answer is:

31 percent

Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48.

To construct an approximate polynomial expression that represents a sequence with constant 4th differences, follow these steps:

Step 1: Understanding Constant Differences

When a sequence has constant \(k\)-th differences, it can be represented by a polynomial of degree \(k\). For example, if:

  • The first differences are calculated (differences of consecutive terms),
  • The second differences (differences of the first differences),
  • The third differences (differences of the second differences),
  • The fourth differences (differences of the third differences),

If the fourth differences are constant, the sequence can be expressed as a polynomial of degree \(4\).

Step 2: Finding the Leading Coefficient

Given that the constant fourth difference is \(48\), we can determine the leading coefficient of the polynomial. The constant \(d\) of the \(n\)-th difference relates to the leading coefficient \(a_4\) of the polynomial \(P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0\) by the formula:

\[ d = 24a_k, \]

where \(d\) is the constant \(k\)-th difference (here \(d = 48\)), and \(a_k\) is the leading coefficient of the polynomial of degree \(k\).

To find \(a_4\):

\[ 48 = 24a_4 \implies a_4 = \frac{48}{24} = 2. \]

Step 3: Setting Up the Polynomial

Now we know the leading coefficient \(a_4 = 2\). The polynomial will have the form:

\[ P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0. \]

Step 4: Determine Remaining Coefficients

To determine \(a_3\), \(a_2\), \(a_1\), and \(a_0\), you would typically need additional information, such as initial terms of the sequence. However, if such values are not provided, you can assume a form or pattern for these coefficients based on the context or the specific behavior of the sequence.

Example Coefficients Assignment

If we assume some arbitrary values for the coefficients based on a particular pattern or context, we might let:

  • \(a_3 = 0\)
  • \(a_2 = 0\)
  • \(a_1 = 0\)
  • \(a_0 = 0\)

In this case, the approximate polynomial expression representing the sequence is:

\[ P(n) = 2n^4. \]

Conclusion

In conclusion, to construct a polynomial expression for a sequence with constant 4th differences of \(48\):

  1. Confirm that the polynomial degree is \(4\).
  2. Use the constant difference to find the leading coefficient \(a_4\).
  3. Construct the polynomial based on the leading coefficient and any available sequence values or assumptions for the lower-degree coefficients.

The final polynomial expression might look like \(P(n) = 2n^4 + C\), where \(C\) can be further refined with more specific information about the sequence terms.