To Win At Lotto In One State, One Must Correctly Select 6 Nu

To Win At Lotto In One State One Must Correctly Select 6 Numbers

1. To win at LOTTO in one state, one must correctly select 6 numbers from a collection of 54 numbers (1 through 54). The order in which the selection is made does not matter. How many different selections are possible? There are ___ different LOTTO selections.

2. In how many ways can a committee of three men and three women be formed from a group of eight men and ten women? A committee of three men and three women can be formed from a group of eight men and ten women in __ different ways.

3. The Senate in a certain state is comprised of 55 Republicans, 41 Democrats, and 4 Independents. How many committees can be formed if each committee must have 3 Republicans and 2 Democrats? ____ committees can be formed.

4. You are dealt one card from a standard 52-card deck. Find the probability of being dealt a diamond. The probability of being dealt diamond is___?

5. You are dealt one card from a standard 52-card deck. Find the probability of being dealt the two of spades. The probability of being dealt the two of spades is?

6. You are dealt one card from a standard 52-card deck. Find the probability of being dealt a heart and a spade. The probability of being dealt a heart and a spade is?

7. A fair coin is tossed three times in succession. The set of equally likely outcomes is (HHH,HHT,HTH,THH,HTT,THT,TTH,TTT). Find the probability of getting exactly two heads is___?

8. A fair coin is tossed 2 times in succession. The set of equally likely outcomes is (HH, HT, TH, TT). Find the probability of getting a tail on the second toss. The probability of getting a tail on the second toss is____?

9. You select a family with three children. If M represents a male child, and F represents a female child, the set of equally likely outcomes for the children’s genders is (MMM,MMF, MFM, MFF, FMM, FMF, FFM, FFF). Find the probability of selecting a family with fewer than 5 male children. P(fewer than 5 male children)=

10. A restaurant offers 8 appetizers and 6 main courses. In how many ways can a person order a two-course meal? There are ___ ways a person can order a two-course meal.

11. A popular brand of pen is available in 3 colors and 2 writing tips. How many different choices of pens do you have with this brand? There are ___ different choices of pens with this brand.

12. An ice cream store sells 5 drinks, in 4 sizes, and 8 flavors. In how many ways can a customer order a drink? There are ___ ways that the customer can order a drink.

13. A restaurant offers a limited lunch menu with main courses (Beef, Pork Roast, Duck, Quiche), vegetables (Broccoli, Carrots, Potatoes), beverages (Coffee, Tea, Milk, Soda), and desserts (Cake, Pie, Sherbet). If one item is selected from each of the four groups, in how many ways can a meal be ordered? There are ___ ways a meal can be ordered.

14. A person can order a new car with options including 8 colors, with or without air conditioning, with or without automatic transmission, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered with regard to these options? There are ___ different ways that a new car can be ordered.

15. You are taking a multiple-choice test with 7 questions. Each question has 4 answer choices, with one correct answer per question. If you select one choice for each question, in how many ways can you answer the questions? You can answer the questions in ___ways.

16. License plates in a particular state display 3 letters followed by 2 numbers. How many different license plates can be manufactured for this state? There are ___ different license plates that can be manufactured for this state.

17. A stock can go up, go down, or stay unchanged. How many possibilities are there if you own 3 stocks? There are ___possibilities with 3 stocks.

18. A 12-sided die is rolled. The set of equally likely outcomes is (1,2,3,4,5,6,7,8,9,10,11,12). Find the probability of rolling a 7. The probability of rolling a 7 is?

19. A 12-sided die is rolled. The set of equally likely outcomes is (1,2,3,4,5,6,7,8,9,10,11,12). Find the probability of rolling a number less than 10. The probability of rolling a number less than 10 is?

20. A 12-sided die is rolled. The set of equally likely outcomes is (1,2,3,4,5,6,7,8,9,10,11,12). Find the probability of rolling a number greater than 2. The probability of rolling a number greater than 2 is?

21. A sales representative can take one of 3 different routes from City A to City E and any one of 6 different routes from City E to City M. How many different routes can she take from City A to City M, going through City E? There are ____ possible routes.

22. Simone, Tyrone, Katrina, Dawn, Ian, and Jim have all been invited to a dinner party. They arrive randomly and each person arrives at a different time. a. In how many ways can they arrive? b. In how many ways can Simone arrive first and Jim last? c. Find the probability that Simone will arrive first and Jim last?

23. A group consists of seven men and six women. Three people are selected to attend a conference. a. In how many ways can three people be selected from this group of thirteen? b. In how many ways can three women be selected from the six women? c. Find the probability that the selected group will consist of all women.

24. To play a certain lottery, a person has to correctly select 6 out of 60 numbers, paying $1 for each six-number selection. If the six numbers picked are the same as the ones drawn by the lottery, mountains of money are bestowed. a. What is the probability that a person with one combination of six numbers will win? b. What is the probability of winning if 100 different lottery tickets are purchased?

25. A box contains 26 transistors, 6 of which are defective. If 6 are selected at random, find the probability that: a. All are defective. b. None are defective.

26. A city council consists of eight Democrats and seven Republicans. If a committee of six people is selected, find the probability of selecting two Democrats and four Republicans.

27. If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability that all 4 cards are diamonds. The probability is?

28. If you are dealt 6 cards from a shuffled deck of 52 cards, find the probability of getting three queens and three kings.

29. You are dealt one card from a 52-card deck. Find the probability that you are not dealt a four. The probability is?

30. You are dealt one card from a 52-card deck. Find the probability that you are not dealt a diamond. The probability is?

31. A single die is rolled. Find the probability of rolling an odd number or a number less than 4. The probability is?

32. You are dealt one card from a 52-card deck. Find the probability that you are dealt a six or a black card. The probability is?

33. The winner of a raffle will receive a 21-foot outboard boat. If 4000 raffle tickets were sold and you purchased 30 tickets, what are the odds against your winning the boat? The odds against winning the boat are?

34. Of the 47 plays attributed to a playwright, 17 are comedies, 10 are tragedies, and 20 are histories. If one play is selected at random, find the odds in favor of selecting a comedy or a tragedy. The odds in favor are?

35. Six stand-up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that: a. Comic E will perform fifth. b. Comic B will perform fifth and comic D will perform first. c. The comedians will perform in the following order: E, B, D, F, C, A. d. Comic C or Comic A will perform second.

Paper For Above instruction

Probability and combinatorial analysis are fundamental areas within mathematics, often applied to diverse fields such as statistics, economics, and computer science to evaluate different outcomes and calculate likelihoods of events. This comprehensive paper addresses a series of probability and combinatorial problems, providing solutions grounded in fundamental principles such as permutations, combinations, and basic probability rules.

1. Calculating the Number of Lotto Selections

The problem involves selecting 6 numbers from a set of 54, where order does not matter. The number of combinations is calculated using the binomial coefficient n choose k:

Number of selections = C(54,6) = 54! / (6! * (54-6)!)

Using factorial calculations, this yields:

C(54,6) = 25,827,165

This means there are 25,827,165 possible Lotto selections.

2. Forming a Committee of Men and Women

To form a committee of three men and three women from 8 men and 10 women, calculate the combinations separately and then multiply:

Number of ways = C(8,3) C(10,3) = (8! / (3! 5!)) (10! / (3! 7!))

= 56 * 120 = 6,720

Thus, there are 6,720 different ways to form such a committee.

3. Forming a Senate Committee

Selection of 3 Republicans out of 55 and 2 Democrats out of 41 yields:

Number of committees = C(55,3) C(41,2) = (55! / (3! 52!)) (41! / (2! 39!))

= 26,235 * 820 = 21,532,950

Approximately 21.53 million committees are possible under these constraints.

4. Probability of Dealing a Diamond

There are 13 diamonds in a standard deck of 52 cards, so:

Probability = 13/52 = 1/4 = 0.25 or 25%.

5. Probability of Dealing the Two of Spades

There is only one two of spades in the deck, so:

Probability = 1/52 ≈ 0.0192 or 1.92%.

6. Probability of Drawing a Heart and a Spade

Since only one card is dealt, these two events are mutually exclusive; you cannot get both in one draw. If the question means "probability of drawing either a heart or a spade", it is:

Number of favorable outcomes = 13 hearts + 13 spades = 26

Probability = 26/52 = 1/2 = 0.5 or 50%.

7. Probability of Exactly Two Heads in Three Tosses

Number of outcomes with exactly two heads = C(3,2) = 3; the outcomes are HHT, HTH, THH. Out of 8 total outcomes, the probability is:

3/8 = 0.375 or 37.5%.

8. Probability of a Tail on the Second Toss in Two Tosses

Possible outcomes: HH, HT, TH, TT. The outcomes with a tail on the second toss are HT and TT:

Number of favorable outcomes = 2

Total outcomes = 4

Probability = 2/4 = 1/2 = 0.5 or 50%.

9. Probability of Fewer than 5 Male Children

Possible numbers of male children: 0 to 3 (since only 3 children). All outcomes are equally likely with probability 1/8 each:

Given the total outcomes: (MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF)

Fewer than 5 males includes all outcomes except, if the total children were more, but since only three children are involved, the probability of fewer than 3 males is:

All outcomes have at most 3 males; hence, the probability that a family has less than 3 males is 1, because total children are only 3.

10. Number of Ways to Order a Two-Course Meal

Number of options for first course = 8 (appetizers), for second course = 6 (main courses), total includes permutations:

Number of arrangements = 8 * 6 = 48

11. Different Choices of Pens

Colors: 3, Tips:2, total options = 3 * 2= 6

12. Ways to Order a Drink

Drinks: 5, Sizes: 4, Flavors: 8, total = 5 4 8 = 160

13. Meal Combinations from Menu

Number of choices = Main courses (4) Vegetables (3) Beverages (4) Desserts (3) = 4 3 4 3 = 144

14. Car Configuration Options

Options: Colors (8), Air conditioning (2), Transmission (2), Power windows (2), CD player (2):

Total configurations = 8 2 2 2 2= 8 * 16 = 128

15. Number of Ways to Answer Test

Each of the 7 questions has 4 choices: total = 4^7 = 16,384

16. License Plates

Letters: 26 choices each, numbers: 10 choices each, total = 26^3 10^2 = 17,576 100 = 1,757,600

17. Possibilities for 3 Stocks

Each stock has 3 possible states (up, down, unchanged): total = 3^3= 27

18. Rolling a 7 on a 12-sided Die

Probability = 1/12 ≈ 0.0833 or 8.33%

19. Rolling a Number Less Than 10

Numbers less than 10: 1 to 9, total 9 outcomes, so probability = 9/12 = 3/4 = 0.75 or 75%

20. Rolling a Number Greater Than 2

Numbers greater than 2: 3-12, total 10 outcomes, probability = 10/12 = 5/6 ≈ 0.8333 or 83.33%

21. Routes with Multiple legs

Routes from A to E: 3, from E to M: 6, total = 3 * 6= 18

22. Arrivals at Dinner Party

a. Total arrangements: 6! = 720. b. Fixing Simone first and Jim last: 4! =24. c. Probability(Simone first, Jim last) = 24/720 = 1/30.

23. Selecting a Group

a. Total selection: C(13,3)= 286. b. Selecting 3 women: C(6,3)= 20. c. Probability all women = 20/286 ≈ 0.0699 or 6.99%.

24. Lottery Winning Probability

a. Probability of matching winning numbers = 1 / C(60,6) ≈ 1 / 50,063,860. b. Buying 100 tickets multiplies the probability: 100 / C(60,6) ≈ 100 / 50,063,860.

25. Transistor Defects

a. All defective: C(6,6) * C(20,0)/ C(26,6) ≈ 1 / 230,230. b. None defective: C(20,6)/ C(26,6) ≈ 38760 / 230230 ≈ 0.1682.

26. Political Committee

Probability of 2 Democrats and 4 Republicans: (C(8,