Seminar problem

Problem

Eight girls and six boys, among them Anna and Tobias, participate in a seminar. For a presentation, a team of four persons is formed by drawing from the participants at random.

1. For each of the following events, give an expression allowing to compute the respective probability.

   \(A\): “Anna and Tobias are in the team.”

   \(B\): “The team consists of the same number of boys and girls.”

2. Describe an event in this context which has the probability represented by the following expression:

\[\frac{\binom{14}{4}-\binom{6}{4}}{\binom{14}{4}}\]

Solution of part a

The combinatorial problem of forming a team of four persons from 14 participants, corresponds to drawing 4 balls from 14 without replacement and disregarding the order. Accordingly, there are

\[\binom{14}{4} = 1001\]

possibilities to form a team.

Because in event \(A\), the team members besides Anna and Tobias are arbitrary, there are \(\binom{12}{2}=66\) possibilities to realize that event.

Accordingly, the probability for event \(A\) is

\[P(A) = \frac{66}{1001} \approx 6.6\%\,.\]

We can check this value by means of a simulation with Sage. In doing so, 4 elements from the numbers 1 to 14 are drawn and Anna and Tobias are assigned to the values \(1\) and \(2\), respectively.

When realizing event \(B\), there are \(\binom{8}{2}=28\) different possibilities to choose two girls and \(\binom{6}{2}=15\) for the boys.

Together there are thus \(28\cdot 15 = 420\) possibilities to form a team of two girls and two boys. The corresponding probability is

\[P(B) = \frac{420}{1001} \approx 42.0\%\,.\]

As before, this result can be checked by means of a simulation. The girls are assigned to the numbers smaller or equal to \(8\) and the numbers above \(8\) correspond to boys.

Solution of part b

The given probability can be simplified to

\[1-\frac{\binom{6}{4}}{\binom{14}{4}}\,.\]

The corresponding event is hence complementary to an event with the probability

\[\frac{\binom{6}{4}}{\binom{14}{4}}\,.\]

The latter corresponds for example to the event “The team contains only boys.” because the number of possibilities to choose 4 boys equals \(\binom{6}{4}\). The complementary event to this then is “The team has at least one girl.”

The second simulation from part a can be easily ajusted to check our interpretation.