Probability #6 - Questions and Answers

Question 1 A bin contains seven red chips, nine green chips, three yellow chips, and six blue chips. Find each probability. drawing a yellow chip, replacing it, and choosing a blue chip. Answer The probability of drawing a yellow chip is 3 / (7+9+3+6) = 3/25 Since there is replacement, the sample space remains the same as 25 chips. So, the probability of drawing a blue chip in the next draw is 6 / (7+9+3+6) = 6/25 So, the probability of drawing a yellow then a blue (with replacement) is: (3/25) * (6/25) = 18/625 Question 2 Triangle XYZ undergoes a single transformation to form the image, triangle X'Y'Z'. The image is congruent to the original figure. Which of the following best describes the possible transformation that took place? O A. a reflection only O B. a rotation only O C. a translation only O D. a reflection, rotation, or a translation Answer The picture goes through a congruent transformation, which might be a reflection, a rotation, or a translation, because it is congruent to the original figure. The solution is D, thus. Question 3 Based on the Normal model N (100, 16) describing IQ scores, what percent of peoples Answer 1 The distribution of IQ scores is normally distributed with a mean of 100 and a standard deviation of 16. X ~ N(100, 16) Transform the normal random variable, X, into the standard normal variable Z by using the following formula: Z = (X - μ) / σ a) P(X > 80) = P((X - 100) / 16 > (80 - 100) / 16) = P(Z > -1.25) = 1 - P(Z < -1.25) = 1 − 0.1056 = 0.8944 = 89.44 b) P(X < 90) = P((X - 100) / 16 < (90 - 100) / 16) = P(Z < -0.625) = 0.2659 = 26.59 c) P(112 < X < 132) = P((112 - 100) / 16 < (X - 100) / 16 < (132 - 100) / 16) = P(0.75 < Z < 2) = P(Z < 2) - P(Z < 0.75) = 0.9772 - 0.7734 = 0.2038 = 20.38 Answer 2 The percentage needed is for the area shown here : Let us suppose the number corresponding to the needed percentage is X, we get the following expression: P(X > 80) Knowing that Z = (X - μ) / σ So, in the next steps, we will get the Z value from X and calculate its percentage as follows: P(X > 80) = P((X - μ) / σ > (80 - μ) / σ) = P(Z > (80 - 100) / 16) = P(Z > -1.25) = 1 − P(Z < -1.25) = 1 - 0.1056 = 0.8944 Therefore, 89.44% of people's IQs would be over 80 b) The percentage to be calculated is for the following colored area: P(X < 90) = P(Z < (90 - 100) / 16) = P(Z < -0.625) = 0.266 Therefore, 26.6% of people's IQs would be under 80 c) The percentage to be calculated represents the data within the covered area: P(112 < X < 132) = P((112 - 100) / 16 < Z < (132 - 100) / 16) = P(0.75 < Z < 2) = 0.2039 Therefore, 20.39% of people's IQs would be between 112 and 132 Answer 3 Step 1. To find the percentage of people with IQ scores below 90, we can calculate the cumulative distribution function (CDF) at 90 using the Normal distribution formula: P(X ≤ 90) = Φ((90 - 100) / 16) Plugging in the values, we get: P(X ≤ 90) = Φ(-10 / 4) = Φ(-2.5) Using a standard Normal distribution table or calculator, we find that Φ(-2.5) ≈ 0.0062. Therefore, the percentage of people with IQ scores below 90 is approximately 0.62%. Step 2. To find the percentage of people with IQ scores above 120, we can calculate the complementary cumulative distribution function (CCDF) at 120: P(X > 120) = 1 - Φ((120 - 100) / 16) Plugging in the values, we get: P(X > 120) = 1 - Φ(20 / 16) = 1 - Φ(5 / 4) = 1 - Φ(1.25) Using a standard Normal distribution table or calculator, we find that Φ(1.25) ≈ 0.8944. Therefore, the percentage of people with IQ scores above 120 is approximately 1 - 0.8944 = 0.1056 Step 3. Finally, to find the percentage of people with IQ scores between 90 and 120, we can subtract the percentages from steps 1 and 2 from 100%: P(90 < X < 120) = 100% - 0.62% - 10.56% = 88.82% P(90 < X < 120) = 100% - 0.62% - 10.56% = 99.38 Therefore, the percentage of people with IQ scores between 90 and 120 is approximately 99.38%. In summary, the percentages of people falling within certain IQ score ranges are approximately 0.62% below 90, 99.38% between 90 and 120, and 0% above 120. Answer 4 The percentage of people with IQ scores below 80 is approximately 10.56%. The percentage of people with IQ scores between 80 and 120 is approximately 78.88%. The percentage of people with IQ scores above 120 is approximately 10.56%. Explanation: Let's find the percentage of people with IQ scores below 80 and above 120. To find the percentage below 80, we need to calculate the cumulative probability up to that point. We can use the cumulative distribution function (CDF) of the Normal distribution: P(X ≤ 80) = Φ((80 - 100) / 16) Where Φ represents the CDF of the Normal distribution. Using this formula, we can find the cumulative probability: P(X ≤ 80) = Φ(-1.25) = 0.1056 Therefore, approximately 10.56% of people have IQ scores below 80. To find the percentage above 120, we can subtract the cumulative probability below 120 from 1: P(X > 120) = 1 - P(X ≤ 120) = 1 - Φ((120 - 100) / 16) Calculating this probability gives us: P(X > 120) = 1 - Φ(1.25) = 1 - 0.8944 = 0.1056 Therefore, approximately 10.56% of people have IQ scores above 120. Now, let's calculate the percentage of people within the range of 80 to 120. To do this, we can subtract the cumulative probabilities below 80 and above 120 from 1: P(80 < X < 120) = 1 - P(X ≤ 80) - P(X > 120) Substituting the values we calculated earlier: P(80 < X < 120) = 1 - 0.1056 - 0.1056 = 0.7888 Therefore, approximately 78.88% of people have IQ scores between 80 and 120. Question 4 The probability density function of the net weight in pounds of a packaged chemical herbicide is f(x) = 2.0 for 49.75 < x < 50.25 pounds. a. Determine the probability that a package weighs more than 50 pounds. b. How much chemical is contained in 90% of all packages? Answer The probability density function f of the random variable X is given as: f(x) = 2, 49.75 < x < 50.25 a) Straightforward calculation gives: P(X > 50) = ∫(from 50 to 50.25) 2 dx = 2 * (50.25 - 50) = 2 * 0.25 = 0.5 b) Write the equality: 0.9 = P(X > x) = ∫(from x to 50.25) 2dy = 2 * (50.25 - x) and solve for x: 0.9 = 2 * (50.25 - x) 0.45 = 50.25 - x x = 50.25 - 0.45 = 49.8 Result: a) 0.5 b) 49.8 Question 5 A closet contains 10 pairs of shoes. If 8 shoes are randomly selected, what is the probability that there will be (a) no complete pair? (b) exactly 1 complete pair? Answer 1 (α) Since there shouldn't be a complete pair, we are allowed to choose at most 1 shoe from each pair. Therefore, our choice can be made in the following way: first choose 8 pairs from the 10 pairs in the closet. this can be done in (10 choose 8) ways. Then from chosen 8 pairs, choose one shoe from each. This can be done in 2^8 ways as there are 2 choices for each pair. Therefore, total number of choices is (10 choose 8) * 2^8. The number of ways to randomly choose 8 shoes out of 20 is (20 choose 8). Therefore, P(no complete pair) = ((10 choose 8) * 2^8) / (20 choose 8) (b) For getting exactly one pair, we can first choose the pair which will appear completely. there are 10 ways of doing it. Then we need to choose 6 shoes from 9 remaining pairs. Proceeding as part(a) we see that there are ((9 choose 6)) * 2^6 ways for this. Therefore, the total number is 10 * ((9 choose 6) * 2^6). Thus, P(exactly one pair) = (10 * ((9 choose 6) * 2^6)) / (20 choose 8) Result: P(no complete pair) = ((10 choose 8) * 2^8) / (20 choose 8) P(exactly one pair) = (10 * ((9 choose 6) * 2^6)) / (20 choose 8) Answer 2 Step 1: (a) To find the probability of selecting no complete pair when 8 shoes are randomly selected, we need to calculate the number of favorable outcomes (no complete pair) and divide it by the total number of possible outcomes. The total number of ways to select 8 shoes out of 20 (10 pairs) is given by (20 choose 8) The number of ways to select 8 shoes with no complete pair can be calculated as follows: Choose 8 shoes from 10 pairs: (10 choose 8) Multiply by 2 for each pair (since we can choose either the left or right shoe from each pair): 2^8 Therefore, the number of favorable outcomes is (10 choose 8) * 2^8 The probability of no complete pair is given by: P({no complete pair}) = ((10 choose 8) * 2^8) / (20 choose 8) Step 2: (b) To find the probability of exactly 1 complete pair when 8 shoes are randomly selected, we again need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. The number of ways to select 1 complete pair from the 10 available pairs is (10 choose 1) Once we have selected a complete pair, we need to choose 6 more shoes from the remaining 18 shoes (9 pairs minus the selected pair). This can be done in (18 choose 6) ways. For each selected pair, we can choose the left or right shoe, so we multiply by 2. Therefore, the number of favorable outcomes is (10 choose 1) * (18 choose 6) * 2^6 The probability of exactly 1 complete pair is given by: P({exactly 1 complete pair}) = ((10 choose 1) * (18 choose 6) * 2^6 ) / (20 choose 8) Answer 3 (a) To find the probability that there will be no complete pair when 8 shoes are randomly selected from 10 pairs, we need to calculate the number of ways this can happen and divide it by the total number of possible outcomes. The total number of ways to select 8 shoes from 20 (10 pairs) is given by the binomial coefficient: (20 choose 8) The number of ways to select 8 shoes without any complete pair is the same as selecting 8 shoes from 18 individual shoes (9 pairs minus 1 complete pair), which can be calculated as: (18 choose 8) Therefore, the probability of no complete pair is: P({no complete pair}) = (18 choose 8) / (20 choose 8) (b) To find the probability of exactly 1 complete pair, we need to consider two cases: Case 1: Selecting 1 complete pair and 6 individual shoes. The number of ways to select 1 complete pair is: (10 choose 1) And the number of ways to select 6 individual shoes from the remaining 18 is: (18 choose 6) Case 2: Selecting 2 complete pairs and 4 individual shoes. The number of ways to select 2 complete pairs is: (10 choose 2) And the number of ways to select 4 individual shoes from the remaining 16 is: (16 choose 4) The total number of favorable outcomes for exactly 1 complete pair is the sum of the two cases: {Favorable outcomes} = ((10 choose 1) * (18 choose 6)) + ((10 choose 2) * (16 choose 4)) Therefore, the probability of exactly 1 complete pair is: P({exactly 1 complete pair}) = (Favorable outcomes) / (20 choose 8)