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Introducing Probability BPS chapter 10 © 2006 W.H. Freeman and Company Probability definition A correct interpretation of the statement “The probability that a child delivered in a certain hospital is a girl is 0.50” would be which one of the following? a) b) c) d) Over a long period of time, there will be equal proportions of boys and girls born at that hospital. In the next two births at that hospital, there will be exactly one boy and one girl. To make sure that a couple has two girls and two boys at that hospital, they only need to have four children. A computer simulation of 100 births for that hospital would produce exactly 50 girls and 50 boys. Probability definition (answer) A correct interpretation of the statement “The probability that a child delivered in a certain hospital is a girl is 0.50” would be which one of the following? a) b) c) d) Over a long period of time, there will be equal proportions of boys and girls born at that hospital. In the next two births at that hospital, there will be exactly one boy and one girl. To make sure that a couple has two girls and two boys at that hospital, they only need to have four children. A computer simulation of 100 births for that hospital would produce exactly 50 girls and 50 boys. Probability From a computer simulation of rolling a fair die ten times, the following data were collected on the showing face: What is a correct conclusion to make about the next ten rolls of the same die? a) b) c) d) The probability of rolling a 5 is greater than the probability of rolling anything else. Each face has exactly the same probability of being rolled. We will see exactly three faces showing a 1 since it is what we saw in the first experiment. The probability of rolling a 4 is 0, and therefore we will not roll it in the next ten rolls. Probability (answer) From a computer simulation of rolling a fair die ten times, the following data were collected on the showing face: What is a correct conclusion to make about the next ten rolls of the same die? a) b) c) d) The probability of rolling a 5 is greater than the probability of rolling anything else. Each face has exactly the same probability of being rolled. We will see exactly three faces showing a 1 since it is what we saw in the first experiment. The probability of rolling a 4 is 0, and therefore we will not roll it in the next ten rolls. Random phenomena Which of the following events would NOT be considered a random phenomenon? a) b) c) d) The event that the next passing car will be blue. The event that a student gets an answer correct after hours of studying. The event that a person’s height is bigger than their armspan. The event that the next customer at a grocery store buys bananas. Random phenomena (answer) Which of the following events would NOT be considered a random phenomenon? a) b) c) d) The event that the next passing car will be blue. The event that a student gets an answer correct after hours of studying. The event that a person’s height is bigger than their armspan. The event that the next customer at a grocery store buys bananas. Probability models If a couple has three children, let X represent the number of girls. Does the table below show a correct probability model for X? a) b) c) d) No, because there are other values that X could be. No, because it is not possible for X to be equal to 0. Yes, because all combinations of children are represented. Yes, because all probabilities are between 0 and 1 and they sum to 1. Probability models (answer) If a couple has three children, let X represent the number of girls. Does the table below show a correct probability model for X? a) b) c) d) No, because there are other values that X could be. No, because it is not possible for X to be equal to 0. Yes, because all combinations of children are represented. Yes, because all probabilities are between 0 and 1 and they sum to 1. Probability If a couple has three children, let X represent the number of girls. What is the probability that the couple does NOT have girls for all three children? a) b) c) d) 0.125 0.125 0.125 + 0.375 = 0.500 1 – 0.125 = 0.825 Probability (answer) If a couple has three children, let X represent the number of girls. What is the probability that the couple does NOT have girls for all three children? a) b) c) d) 0.125 0.125 0.125 + 0.375 = 0.500 1 – 0.125 = 0.825 Probability If a couple has three children, let X represent the number of girls. What is the probability that the couple has either one or two girls? a) b) c) d) 0.375 0.375 + 0.375 = 0.750 1 - 0.125 = 0.825 0.500 Probability (answer) If a couple has three children, let X represent the number of girls. What is the probability that the couple has either one or two girls? a) b) c) d) 0.375 0.375 + 0.375 = 0.750 1 - 0.125 = 0.825 0.500 Density curves A random number generator was used to generate random numbers along the interval from -2 to +2. The density curve of the generated data is shown below. What proportion of values will lie between -1 and +2? a) b) c) d) 1.00 0.50 0.25 0.75 Density curve A random number generator was used to generate random numbers along the interval from 1 to 5. The density curve of the generated data is shown below. At what value, y, must the blue line be placed in order to have 25% of the data between 1 and y? Area = 0.25 a) b) c) d) 1.5 2 2.5 4 Density curve (answer) A random number generator was used to generate random numbers along the interval from 1 to 5. The density curve of the generated data is shown below. At what value, y, must the blue line be placed in order to have 25% of the data between 1 and y? Area = 0.25 a) b) c) d) 1.5 2 2.5 4 Random variables Would the following random variable, X, be discrete or continuous? X = the number of sales at the drive-through during the lunch rush at the local fast food restaurant. a) b) Continuous Discrete Random variables (answer) Would the following random variable, X, be discrete or continuous? X = the number of sales at the drive-through during the lunch rush at the local fast food restaurant. a) b) Continuous Discrete Random variables Would the following random variable, X, be discrete or continuous? X = the time required to run a marathon. a) b) Continuous Discrete Random variables (answer) Would the following random variable, X, be discrete or continuous? X = the time required to run a marathon. a) b) Continuous Discrete Random variables Would the following random variable, X, be discrete or continuous? X = the number of fans in a football stadium. a) b) Continuous Discrete Random variables (answer) Would the following random variable, X, be discrete or continuous? X = the number of fans in a football stadium. a) b) Continuous Discrete Random variables Would the following random variable, X, be discrete or continuous? X = the distance a car could drive with only one gallon of gas. a) b) Continuous Discrete Random variables (answer) Would the following random variable, X, be discrete or continuous? X = the distance a car could drive with only one gallon of gas. a) b) Continuous Discrete