An Unfair Die Suppose an unfair die is rolled and let X be the random variable representing the number of dots that would appear with a probability distribution below. Outcome 2 3 4 5 6 (x) Probability 0.1 0. 1 0.1 0.5 0.1 0.1 P(x) Questions: 1. What will be the average number of dots that would appear? 2. How does the assumed value of the outcome vary from the average number of dots that would appear? 3. Will you join in a game of chance using an unfair die?The number of cellular phones sold per day at the E-Cell Retail Store with the corresponding probabilities is shown in the table below. Compute the mean, variance, and standard deviation and interpret the result. Number of cellphones sold per day in a retail store (x) 15 18 19 20 22 Probability (P(x) 0.30 0.20 0.20 0. 15 0. 15 Solution: Complete the statement: Ux [xi . P(x)] = 15(0,30) + 18(0.20) + 19(0.20) + 20(0.15) + 22(0.15) = _The mean is equal to therefore, it means that the average number of cellular phones of sold per day is To find the variance complete the table below: x2 . P(x ) x P(x) x2 x . P(x) 15 0.30 4.5 18 0.20 3.6 19 0.20 3.8 20 0.15 3 22 0.15 3.3 [[x2 . P ( xx )] = 02 = [(x2 . P( 30)] - 12 = _ 0 = Vo2 = Therefore, the variance of a probability distribution is equal to while the standard deviation is equal to Independent Activity 1 Beth's Bread and Pastry Shop determines the number of cupcakes sold per day with its corresponding probabilities. Find the mean, variance, and standard deviation of the probability distribution below. If Beth, the owner of the shop is claiming that the average number of cupcakes sold in a day is 150 pieces, do you think it is a believable claim? Number of cupcakes sold per day (x) 90 120 135 150 160 175 Probability P(x) 0. 15 0.10 0.20 0.20 0.20 0.15 Practice Activity 2 Analyze the following pairs of data and identify which of the following will most likely yield to a higher variance and higher standard deviation. Put a check mark on the appropriate box. 1. Number of students from different grade levels. Number of boys in a family with three children. 2. Number of fish inside the aquarium from different households. Number of fish inside a can from the different local brands of sardines. 3. Number of COVID-19 patients from different hospitals. Number of family members with fever in a barangay. 4. Number of M & M's peanuts inside a 1.69 Oz bag from different retail stores.0 Number of kernels in corn of different sizes. 5. L Number of players in a group playing the game "the boat is sinking." O Number of passengers in different luxury cruise ships. Independent Activity 2 Give examples of at least two pairs of data that will most likely yield different variance and standard deviation. Identify the one with a higher possible value of variance and explain why.1. Which of the following is an example of a discrete random variable? A. weight of newborn babies B. body temperature of COVID-19 patients C. number of heads that will come out if you toss a coin twice D. height of basketball players 2. Which of the following best describe the mean of a discrete random variable? A. It is the lowest assumed value of a discrete random variable. B. It is the highest assumed value of a discrete random variable. C. It is the average value of a discrete random variable over numerous trials of an experiment. D. It is the amount of spread, dispersion, or variability of the assumed value of a discrete random variable. 3. Which of the following best describe the variance and standard deviation of a probability? A. It is the lowest assumed value of a discrete random variable. B. It is the highest assumed value of a discrete random variable. C. It is the average value of a discrete random variable over numerous trials of an experiment. D. It is the amount of spread, dispersion, or variability of the assumed value of a discrete random variable.4. Which of the following best describe the standard deviation of a probability distribution? A. It is twice the variance. B. It is the product of the mean and the variance. C. It is the ratio of the mean and the variance. D. It is the square root of the variance. 5. How would you interpret a very small variance or standard deviation? B. A. The values of the random variables are equal to the mean. The values of the random variables are closer to the mean. C. The values of the random variables are farther from the mean. D. The values of the random variables have no relationship with the mean. 6. Which of the following data show most likely the largest possible variance or variability? A. number of pieces of French fries in a regular pack from different orders of customers at Mcdonalds B. number of boys in families of three-children C. number of customers per hour who went shopping at SM Super Malls D. number of heads that will appear if two coins are tossed together repeatedly 7. Which of the following data show most likely the smallest possible variance or variability? A. the number of passengers in a tricycle per destinations B. the number of applicants in the different job opening C. the number of families who own a private vehicle in different cities in NCR D. the number of adults who use public restrooms in Metro Manila 8. What formula is described by o = VE[x2P(x)] -u2 ? A. the mean of a discrete random variable B. the variance of a discrete random variable C. the standard deviation of a discrete random variable D. the expected value of a discrete random variable For numbers 9 -12, refer to the probability distribution of the number of books borrowed from a school library in a day and its corresponding probabilities. X 20 25 30 35 40 45 P(x) 0.1 0.1 0.4 0.2 0.1 0.1 9. What is the mean of the probability distribution? A. 25 B. 29 C. 30 D. 32 10. How would you interpret the mean value that you get from item number 8? A. It is the least number of books borrowed from the school library in a day. B. It is the largest number of books borrowed from the school library in a day. C. It is the average number of books borrowed from the school library in a day. D. It is the difference between the largest and the least number of books borrowed from the school library in a day. 11. What is the variance of the probability distribution? A. 38 B. 40 C. 43 D. 46 12. What is the standard deviation of the probability distribution? A. 6.16 B. 6.32 C. 6.56 D. 6.78 13. Which of the following is NOT a property of the variance? A. A small variance means that the distribution of the random variable is narrowly concentrated around the mean. B. A large variance means that the distribution is spread out, with some chance of observing values at some distance from the mean. C. The variance is a value that is always positive. D. The variance is a value that is always negative.For numbers 14-15. The mean of the probability distribution below is equal to 37.05 with a variance of 36.75 and a standard deviation of 6.06. Number of ice candy sold per 45 day in a retail 30 32 36 40 42 store (x) Probability (P(x) 0: 10 0.25 0.30 0.10 0.15 0.10 14. How would you interpret the mean value of 37.05? A. The least number of ice candy that will be sold in a day is 37 pieces. B. The highest number of ice candy that will be sold in a day is 37 pieces. C. The average number of ice candy that will be sold in a day is 37 pieces. D. No interpretation can be made about the mean value of 37.05. 15. If you are the owner of the retail store, how many ice candies will you prepare to ensure that you can supply the demands of your customers every day? A. 10 pieces and below C. 21-30 pieces B. 11-20 pieces D. 30 pieces and above