Questions: Use the percentages of students enrolled in each college at our university to simulate the probability that in a group of 10 students selected at random, at least one other person is in the same major as you. Engineering- 0.1551 Humanities- 0.0693 Communications- 0.1926 Agriculture- 0.0612 Business- 0.1458 Natural Science- 0.1271 Undergrad- 0.0506 Behavioral- 0.2675

Use the percentages of students enrolled in each college at our university to simulate the probability that in a group of 10 students selected at random, at least one other person is in the same major as you. Engineering- 0.1551 Humanities- 0.0693 Communications- 0.1926 Agriculture- 0.0612 Business- 0.1458 Natural Science- 0.1271 Undergrad- 0.0506 Behavioral- 0.2675
Transcript text: Use the percentages of students enrolled in each college at our university to simulate the probability that in a group of 10 students selected at random, at least one other person is in the same major as you. Engineering<- 0.1551 Humanities<- 0.0693 Communications<- 0.1926 Agriculture<- 0.0612 Business<- 0.1458 Natural_Science<- 0.1271 Undergrad<- 0.0506 Behavioral<- 0.2675
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Solution

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Solution Steps

Step 1: Define the Problem

We want to determine the probability that in a group of 10 students selected at random, at least one other student is in the same major as you. This can be calculated using the complementary probability approach.

Step 2: Calculate the Complementary Probability

The probability that no other student is in the same major as you can be expressed as:

\[ P(X = 0) = \binom{n-1}{0} \cdot p^0 \cdot q^{n-1} = q^{n-1} \]

where:

  • \( n = 10 \) (total students),
  • \( p \) is the probability of being in a specific major,
  • \( q = 1 - p \) is the probability of not being in that major.

Thus, the probability that at least one other student is in the same major is:

\[ P(X \geq 1) = 1 - P(X = 0) = 1 - q^{n-1} \]

Step 3: Calculate for Each Major

Using the probabilities for each major, we find:

  1. Engineering:

    • \( p = 0.1551 \)
    • \( q = 0.8449 \)
    • \( P(X \geq 1) = 1 - (0.8449)^{9} \approx 0.7806 \)

  2. Humanities:

    • \( p = 0.0693 \)
    • \( q = 0.9307 \)
    • \( P(X \geq 1) = 1 - (0.9307)^{9} \approx 0.4761 \)

  3. Communications:

    • \( p = 0.1926 \)
    • \( q = 0.8074 \)
    • \( P(X \geq 1) = 1 - (0.8074)^{9} \approx 0.8542 \)

  4. Agriculture:

    • \( p = 0.0612 \)
    • \( q = 0.9388 \)
    • \( P(X \geq 1) = 1 - (0.9388)^{9} \approx 0.4336 \)

  5. Business:

    • \( p = 0.1458 \)
    • \( q = 0.8542 \)
    • \( P(X \geq 1) = 1 - (0.8542)^{9} \approx 0.7579 \)

  6. Natural Science:

    • \( p = 0.1271 \)
    • \( q = 0.8729 \)
    • \( P(X \geq 1) = 1 - (0.8729)^{9} \approx 0.7058 \)

  7. Undergrad:

    • \( p = 0.0506 \)
    • \( q = 0.9494 \)
    • \( P(X \geq 1) = 1 - (0.9494)^{9} \approx 0.3733 \)

  8. Behavioral:

    • \( p = 0.2675 \)
    • \( q = 0.7325 \)
    • \( P(X \geq 1) = 1 - (0.7325)^{9} \approx 0.9393 \)

Final Answer

The probabilities that at least one other student is in the same major as you are as follows:

  • Engineering: \( \approx 0.7806 \)
  • Humanities: \( \approx 0.4761 \)
  • Communications: \( \approx 0.8542 \)
  • Agriculture: \( \approx 0.4336 \)
  • Business: \( \approx 0.7579 \)
  • Natural Science: \( \approx 0.7058 \)
  • Undergrad: \( \approx 0.3733 \)
  • Behavioral: \( \approx 0.9393 \)

Thus, the final boxed answers are:

\[ \boxed{ \begin{align_} \text{Engineering} & : 0.7806 \\ \text{Humanities} & : 0.4761 \\ \text{Communications} & : 0.8542 \\ \text{Agriculture} & : 0.4336 \\ \text{Business} & : 0.7579 \\ \text{Natural Science} & : 0.7058 \\ \text{Undergrad} & : 0.3733 \\ \text{Behavioral} & : 0.9393 \\ \end{align_} } \]

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