1. A teacher has 10 boys and 12 girls in her grade school class. In how many ways can she select 8 of the children to be in a play if she must select 5 boys and 3 girls.
2. The manager of the various displays in the Acme Fine Clothing Emporium has a window which will hold 4 dummies. He has 8 dummies, each showing a different style outfit, from which to choose. How many different displays can he arrange in his display window? (If the order in which the dummies appear changes, it is regarded by the manager as ‘a different display.)
3. If you buy a $300,000 house, paying 5% down, and take out a 25 year, fixed rate of 6.5%, loan, then your monthly payments for principal and interest will be
4. Three coins are tossed. Find the probability that all the coins land heads up
5. Two dice are rolled. Find the probability of getting a sum greater than 10.
6. A person rolls two dice. What are the odds in favor of throwing at least an eight?
7. A lottery has one $2000 prize, two $1000 prizes, and ten $500 prizes. Thirty-six hundred tickets are sold at $4 each. Find the expected value if a person buys two tickets.
8. A card is drawn from a standard 52 card deck. What is the probability that it is either a red ‘ace or a black king?
9. A card is drawn from a standard 52 card deck. What is the probability that it is either a red’ ace or a black king?
10 How many three symbol codes using the digits 2 through 9 are possible if repetitions are allowed? (243 is an example of a three symbol code.)
11. An urn contains 5 red marbles, 4 blue marbles, and 3 green marbles. A marble is selected at random and then, without replacing the first marble, a second marble is selected at random. What is the probability of selecting a green marble and then a red marble?
12. In a classroom, the students are 20 boys and 28 girls. If one student is selected at random, find the probability that the student is a boy.
If you include only a couple of problems in each post, along with your attempts of answers, someone may be able to help you.
If you don't do this, I will assume that you don't want help, just answers.
1. To solve this problem, we need to use the concept of combinations. The number of ways to select 5 boys from 10 is given by the combination formula "nCr" which stands for "n choose r". In this case, we have 10 boys and we want to choose 5, so the formula would be 10C5. Similarly, the number of ways to select 3 girls from 12 is given by 12C3. To find the total number of ways to select 5 boys and 3 girls, we need to multiply the number of ways to select boys by the number of ways to select girls, so the final answer is (10C5) * (12C3).
2. To find the number of different displays the manager can arrange, we need to use the concept of permutations. The number of ways to arrange 8 dummies in a window that can hold 4 is given by the permutation formula "nP r" which stands for "n permute r". In this case, we have 8 dummies and we want to arrange them in a window of size 4, so the formula would be 8P4.
3. To calculate the monthly payments for principal and interest on a loan, we can use the mortgage loan formula. The formula is:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]
Where:
M = Monthly payment
P = Loan amount (in this case, $300,000)
i = Monthly interest rate (6.5% divided by 12 months)
n = Number of payments (25 years multiplied by 12 months per year)
By plugging in the values into the formula, you can calculate the monthly payments.
4. When three coins are tossed, there are 2^3 = 8 possible outcomes (each coin can land heads or tails). Out of these 8 outcomes, there is only 1 outcome where all three coins land heads up. Therefore, the probability that all the coins land heads up is 1/8.
5. The total number of possible outcomes when rolling two dice is 6 * 6 = 36 (each die has 6 possible outcomes). To find the probability of getting a sum greater than 10, we need to count the number of outcomes where the sum is greater than 10. There are 3 outcomes that fit this condition: (5,6), (6,5), and (6,6). Therefore, the probability is 3/36, which simplifies to 1/12.
6. To find the odds in favor of throwing at least an eight when rolling two dice, we need to count the number of favorable outcomes and the number of possible outcomes. There are 5 favorable outcomes: (2,6), (3,5), (4,4), (5,3), and (6,2). The total number of possible outcomes is 6 * 6 = 36. So, the odds in favor of throwing at least an eight are 5 to 31 (or 5:31).
7. To find the expected value, we need to multiply each prize amount by the probability of winning that prize and sum up all the results. The probability of winning the $2000 prize is 1/3600, the probability of winning a $1000 prize is 2/3600, and the probability of winning a $500 prize is 10/3600. Therefore, the expected value is (1/3600 * $2000) + (2/3600 * $1000) + (10/3600 * $500).
8. To find the probability of drawing either a red ace or a black king from a standard deck of 52 cards, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes. There are 2 red aces and 2 black kings. The total number of cards in the deck is 52. Therefore, the probability is (2 + 2) / 52 = 4/52, which simplifies to 1/13.
9. This question seems to be a duplicate of question 8.
10. Since repetitions are allowed, there are 8 possible digits for each symbol in a three-symbol code. Therefore, the total number of possible codes is 8 * 8 * 8 = 512.
11. To find the probability of selecting a green marble and then a red marble, we need to multiply the probability of selecting a green marble by the probability of selecting a red marble after not replacing the first marble. The probability of selecting a green marble first is 3/12 (3 green marbles out of 12 total marbles). After selecting the green marble, there will be 11 marbles left, including 5 red marbles. Therefore, the probability of selecting a red marble second is 5/11. Multiplying these probabilities, we get (3/12) * (5/11).
12. To find the probability of selecting a boy student from the classroom, we need to divide the number of boy students by the total number of students. There are 20 boy students and 28 girl students, so the total number of students is 20 + 28 = 48. Therefore, the probability is 20 / 48, which simplifies to 5/12.