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![Question 6: In a deck of 40 cards, there are 10 red cards, 15 blue cards, and 15 green cards. If 4 cards are drawn
without replacement, what is the probability of drawing 2 red cards and 2 blue cards?
Answer 6: To calculate the probability of drawing 2 red cards and 2 blue cards, we need to use combinations.
The total number of ways to draw 4 cards out of 40 is (40 choose 4). The number of ways to draw 2 red cards
out of 10 is (10 choose 2), and the number of ways to draw 2 blue cards out of 15 is (15 choose 2). Multiply
these combinations and divide by the total combinations: [(10 choose 2) * (15 choose 2)] / (40 choose 4).
Simplify to find the probability.
Question 7: A box contains 6 red balls and 8 black balls. Three balls are drawn with replacement. What is the
probability of getting exactly 2 red balls?
Answer 8: Since the balls are drawn with replacement, each draw is independent and doesn't affect the
probabilities of subsequent draws. The probability of drawing a red ball is 6/14, and the probability of drawing a
black ball is 8/14. To get exactly 2 red balls in 3 draws, we consider the binomial distribution formula: P(X = k) =
(n choose k) * p^k * q^(n-k), where n is the number of trials, k is the number of successful outcomes, p is the
probability of success, and q is the probability of failure. In this case, n = 3, k = 2, p = 6/14, and q = 8/14. Plug
these values into the formula to calculate the probability.](https://image.slidesharecdn.com/staticshh-230814050436-da9b878d/75/Your-Statistics-Homework-Solver-is-Here-5-2048.jpg)
![Question 6: In a deck of 40 cards, there are 10 red cards, 15 blue cards, and 15 green cards. If 4 cards are drawn
without replacement, what is the probability of drawing 2 red cards and 2 blue cards?
Answer 6: To calculate the probability of drawing 2 red cards and 2 blue cards, we need to use combinations.
The total number of ways to draw 4 cards out of 40 is (40 choose 4). The number of ways to draw 2 red cards
out of 10 is (10 choose 2), and the number of ways to draw 2 blue cards out of 15 is (15 choose 2). Multiply
these combinations and divide by the total combinations: [(10 choose 2) * (15 choose 2)] / (40 choose 4).
Simplify to find the probability.
Question 7: A box contains 6 red balls and 8 black balls. Three balls are drawn with replacement. What is the
probability of getting exactly 2 red balls?
Answer 8: Since the balls are drawn with replacement, each draw is independent and doesn't affect the
probabilities of subsequent draws. The probability of drawing a red ball is 6/14, and the probability of drawing a
black ball is 8/14. To get exactly 2 red balls in 3 draws, we consider the binomial distribution formula: P(X = k) =
(n choose k) * p^k * q^(n-k), where n is the number of trials, k is the number of successful outcomes, p is the
probability of success, and q is the probability of failure. In this case, n = 3, k = 2, p = 6/14, and q = 8/14. Plug
these values into the formula to calculate the probability.](https://crownmelresort.com/image.slidesharecdn.com/staticshh-230814050436-da9b878d/75/Your-Statistics-Homework-Solver-is-Here-5-2048.jpg)
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Your Statistics Homework Solver is Here! 📊📚
The document covers various probability problems involving drawing cards, rolling dice, and selecting balls or marbles, emphasizing calculations using combinations and the binomial distribution. Detailed solutions for scenarios such as drawing specific cards from a deck, calculating the outcome of rolling a die, and drawing marbles without replacement highlight the principles of independent events and sample spaces. Each problem is approached by breaking down the probability into manageable parts and employing mathematical formulas for accurate results.
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- 2. Question:In a deckof 52 playing cards, three cards are drawn without replacement. Find the probability that the firs card is a King, the second card is a Queen, and the third card is a Jack. Answer:To solve this problem, we'll break down the probability calculation step by step. 1. Probability of drawing a King as the first card: There are 4 Kings in a deck of 52 cards, so the probability of drawi a King as the first card is 4/52. 2. Probability of drawing a Queen as the second card: After drawing the King, there are 51 cards left in the deck, including 4 Queens. Thus, the probability of drawing a Queen as the second card is 4/51. 3. Probability of drawing a Jack as the third card: After drawing the King and Queen, there are 50 cards left, includin 4 Jacks. So, the probability of drawing a Jack as the third card is 4/50. Since these events are independent (drawing one card doesn't affect the probability of drawing another), we can multiply the probabilities: Total Probability = (4/52) * (4/51) * (4/50). Simplify the fractions and calculate the result. The final probability is the product of these three fractions.
- 3. Question 1: Abox contains 10 red balls and 15 blue balls. Three balls are drawn without replacement. Calculate the probability of drawing 2 red balls followed by 1 blue ball. Answer 1: To find the probability of this sequence, we break it down into steps. The probability of drawing the first red ball is 10/25. After that, the probability of drawing a second red ball, without replacement, is 9/24. Lastly, the probability of drawing a blue ball, out of the remaining 23 balls, is 15/23. Since these events are independent, we multiply these probabilities: (10/25) * (9/24) * (15/23) = 1350 / 13800 = 25 / 256 ≈ 0.0977. Question 2: A fair six-sided die is rolled four times. Find the probability of getting exactly two 3s and two 5s. Answer 2: The probability of rolling a 3 on a fair six-sided die is 1/6, and the probability of rolling a 5 is also 1/6. The probability of getting exactly two 3s and two 5s in four rolls can be calculated using the binomial distribution formula: P(X = k) = (n choose k) * p^k * q^(n-k), where n is the number of trials, k is the number of successful outcomes, p is the probability of success, and q is the probability of failure. In this case, n = 4, k = 2, p = 1/6, and q = 5/6. Plugging these values into the formula gives P(X = 2) = (4 choose 2) * (1/6)^2 * (5/6)^2 = 6 * 1/36 * 25/36 = 150 / 1296 ≈ 0.1157.
- 4. Question 3: Anurn contains 8 red balls and 12 green balls. If 5 balls are drawn without replacement, what is the probability of drawing at least 2 red balls? Answer 3: To find the probability of drawing at least 2 red balls, we consider two cases: drawing exactly 2 red balls and drawing 3 red balls. The probability of drawing exactly 2 red balls can be calculated using combinations: (8 choose 2) * (12 choose 3) / (20 choose 5). Similarly, the probability of drawing 3 red balls is (8 choose 3) * (12 choose 2) / (20 choose 5). Add these two probabilities to get the total probability of drawing at least 2 red balls. Question 4: A spinner is divided into 8 equal sectors, numbered 1 through 8. If the spinner is spun twice, what is the probability that the sum of the two numbers is odd? Answer 4: There are 4 odd and 4 even numbers on the spinner. To find the probability of the sum of two numbers being odd, we consider two cases: odd + odd and even + even. For odd + odd, the probability is (4/8) * (3/8) = 12/64. For even + even, the probability is (4/8) * (3/8) = 12/64. Summing these probabilities gives 24/64 = 3/8. Question 5: A bag contains 5 red marbles, 4 green marbles, and 6 blue marbles. If 3 marbles are drawn without replacement, what is the probability of drawing 2 red marbles and 1 green marble? Answer 5: The probability of drawing 2 red marbles and 1 green marble can be calculated using combinations. The total number of ways to draw 3 marbles out of 15 is (15 choose 3). The number of ways to draw 2 red marbles out of 5 is (5 choose 2), and the number of ways to draw 1 green marble out of 4 is (4 choose 1). Multiply these combinations and divide by the total combinations to get the probability. The result will be (5 choose 2) * (4 choose 1) / (15 choose 3).
- 5. Question 6: Ina deck of 40 cards, there are 10 red cards, 15 blue cards, and 15 green cards. If 4 cards are drawn without replacement, what is the probability of drawing 2 red cards and 2 blue cards? Answer 6: To calculate the probability of drawing 2 red cards and 2 blue cards, we need to use combinations. The total number of ways to draw 4 cards out of 40 is (40 choose 4). The number of ways to draw 2 red cards out of 10 is (10 choose 2), and the number of ways to draw 2 blue cards out of 15 is (15 choose 2). Multiply these combinations and divide by the total combinations: [(10 choose 2) * (15 choose 2)] / (40 choose 4). Simplify to find the probability. Question 7: A box contains 6 red balls and 8 black balls. Three balls are drawn with replacement. What is the probability of getting exactly 2 red balls? Answer 8: Since the balls are drawn with replacement, each draw is independent and doesn't affect the probabilities of subsequent draws. The probability of drawing a red ball is 6/14, and the probability of drawing a black ball is 8/14. To get exactly 2 red balls in 3 draws, we consider the binomial distribution formula: P(X = k) = (n choose k) * p^k * q^(n-k), where n is the number of trials, k is the number of successful outcomes, p is the probability of success, and q is the probability of failure. In this case, n = 3, k = 2, p = 6/14, and q = 8/14. Plug these values into the formula to calculate the probability.
- 6. Question 3: Acompany has 4 software engineers and 6 data analysts. If 3 employees are selected randomly to form a team, what is the probability that the team consists of 2 software engineers and 1 data analyst? Answer 3: To find the probability of forming a team with 2 software engineers and 1 data analyst, we use combinations. The total number of ways to select 3 employees out of 10 is (10 choose 3). The number of ways to select 2 software engineers out of 4 is (4 choose 2), and the number of ways to select 1 data analyst out of 6 is (6 choose 1). Multiply these combinations and divide by the total combinations to get the desired probability. Question 4: A fair coin is tossed 5 times. What is the probability of getting at least 3 heads? Answer 4: To calculate the probability of getting at least 3 heads in 5 coin tosses, we consider the different combinations of successful outcomes: 3 heads, 4 heads, and 5 heads. The probability of getting a head in a fair coin toss is 0.5, and the probability of getting a tail is also 0.5. Using the binomial distribution formula, we calculate the probabilities of these individual cases and then add them up to get the probability of getting at least 3 heads.
- 7.