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Counting teams with Cooper

A team of 3 student volunteers is formed at random from a class of 40. What is the probability that Cooper is on the team?

Haley and Hugo from 2 groups of 3

A UVA class has 40 students. Suppose the professor chooses 3 students on Wednesday at random, and again 3 on Friday. What is the probability that Haley is chosen today and Hugo on Friday?

Solution

(1) Count total outcomes:

  • We have possible groups chosen Wednesday.
  • We have possible groups chosen Friday.

Therefore possible groups in total. (Product of possibilities.)

(2) Count desired outcomes:

The possible groups of 3 that include Haley can be counted by counting the subgroups of 2 formed of the other students in Haley’s group.

  • Therefore we have groups that contain Haley.
  • Similarly, we have groups that contain Hugo.

Therefore we count total desired outcomes.

(3) Compute probability:

Let label the desired event. By the counting rule:

Evaluate using our data:

Counting VA license plates

VA license plates have three letters (with no I, O, or Q) followed by four numerals. A random plate is seen on the road.

(a) What is the probability that the numerals occur in strictly increasing order?

(b) What is the probability that at least one number is repeated?

Solution

(a) Numerals in increasing order.

(1) Count total plates:

  • Have options for letters.
  • Have options for numbers.

Thus possible plates.

(2) Count ways to have 4 numerals that occur in increasing order:

There are ways to choose 4 distinct numerals from 10 options.

For each choice of four distinct numerals, there is exactly one ordering that’s increasing.

Therefore, there are ways to have 4 numerals that occur in increasing order.

(3) Count ways to have a list of 3 letters (excluding I, O, Q).

There are 26 total letters, 3 are excluded, thus 23 options for each letter.

Repetition is allowed, thus we have total ways.

(3) Compute probability:

Total count of desired plates (taking the product of possibilities):

Let label the event that a plate has increasing numerals. The counting formula for probability is:

Evaluate using our data:

(b) At least one number repeated.

“At least” is hard to work with! Lots of ways that can happen.

Try the complement event, which is much simpler: “no repetition”

Let be event that no numbers are repeated. All are distinct. Then is our desired event.

Count the possibilities:

The total number of plates is still .

Therefore, license plates with at least one number repeated:

Desired outcomes over total outcomes:

Counting out 4 teams

A board game requires 4 teams of players. How many configurations of teams are there out of a total of 17 players if the number of players per team is 4, 4, 4, 5, respectively.