Probability and Combinatorics Quiz

A lunch menu consists of 5 different kinds of sandwiches, 2 different kinds of soup, and 3 different drinks. How many choices are there for ordering a sandwich, a bowl of soup, and a drink?

If twelve people are entered in a race, in how many ways can the first three places come out?

A committee is to consist of two members. If there are five men and six women available to serve on the committee, how many different committees can be formed?

A number cube is rolled 400 times with the following results: 63 ones, 68 twos, 57 threes, 66 fours, 65 fives, and 81 sixes. What is the experimental probability of rolling an even number?

A drawer contains 10 red socks, 9 white socks, and 8 blue socks. What is the probability that the first sock and the second sock drawn are both white?

Which of these pairs of events are dependent?

Which of these pairs of events are independent?

Find the number of distinguishable permutations of the letters MOVIE.

Lunch menu includes 5 sandwich types, 2 soup types, and 3 drink types.
Total choices for ordering one of each item: 5 × 2 × 3 = 30 options.

Race Outcomes

Twelve participants in a race with no ties to determine outcomes.
The number of ways to arrange the first three places: 12P3 = 12!/(12-3)! = 2640.

Committee Formation

A committee consists of 2 members selected from 5 men and 6 women.
The number of different committees that can be formed is calculated using combinations: C(11, 2) = 165.

Experimental Probability

A cube rolled 400 times with recorded results:
- 63 ones, 68 twos, 57 threes, 66 fours, 65 fives, and 81 sixes.
Total even outcomes (2, 4, 6): 68 + 66 + 81 = 215.
Experimental probability of rolling an even number: 215/400 = 0.64.

Probability of Drawing Socks

A drawer contains 10 red, 9 white, and 8 blue socks.
Drawing two white socks without replacement:
- Probability of first sock being white: 9/27.
- Probability of second sock being white after the first: 8/26.
- Combined probability = (9/27) × (8/26) = 4/75.

Dependent Events

Flipping a coin twice creates dependent outcomes.
The result of the first flip influences the possible outcomes for the second flip.

Independent Events

Drawing names with replacement exemplifies independent events.
The outcome of the first draw does not affect the second draw.

Distinguishable Permutations

The word "MOVIE" consists of 5 unique letters.
The number of distinguishable permutations is calculated as 5! = 120.

Test your understanding of probability and combinatorial math with this quiz. It covers various topics, including lunch menu choices, race outcomes, committee formation, and experimental probability. Perfect for students looking to strengthen their skills in these areas.