Page 118 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
2. Understanding Probability Distributions for Continuous Random Variables
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
1.5 recognize conditions (e.g., dependent trials) that give rise to a random variable that fol- lows a hypergeometric probability distribu- tion, calculate the probability associated with each value of the random variable (e.g., by using a tree diagram; by using combinations), and represent the distribution numerically using a table and graphically using a proba- bility histogram
1.6 compare, with technology and using numeric and graphical representations, the probability distributions of discrete random variables (e.g., compare binomial distributions with the same probability of success for increasing numbers of trials; compare the shapes of a hypergeometric distribution and a binomial distribution)
Sample problem: Compare the probability distributions associated with drawing 0, 1, 2, or 3 face cards when a card is drawn 3 times from a standard deck with replacement (i.e., the card is replaced after each draw) and without replacement (i.e., the card is not replaced after each draw).
1.7 solve problems involving probability distri- butions (e.g., uniform, binomial, hypergeo- metric), including problems arising from real-world applications
Sample problem: The probability of a busi- ness person cancelling a reservation at La Place Pascal hotel is estimated to be 8%. Generate and graph the probability distribu- tion for the discrete random variable that represents the number of business people cancelling when there are 10 reservations. Use the probability distribution to determine the probability of at least 4 of the 10 reserva- tions being cancelled.
By the end of this course, students will:
2.1 recognize and identify a continuous random variable (i.e., a variable that assumes values from the infinite number of possible outcomes in a continuous sample space), and distinguish between situations that give rise to discrete frequency distributions (e.g., counting the number of outcomes for drawing a card or tossing three coins) and situations that give rise to continuous frequency distributions
(e.g., measuring the time taken to complete a task or the maximum distance a ball can be thrown)
2.2 recognize standard deviation as a measure
of the spread of a distribution, and determine, with and without technology, the mean and standard deviation of a sample of values of
a continuous random variable
2.3 describe challenges associated with determin- ing a continuous frequency distribution (e.g., the inability to capture all values of the vari- able, resulting in a need to sample; uncer- tainties in measured values of the variable), and recognize the need for mathematical models to represent continuous frequency distributions
2.4 represent, using intervals, a sample of values of a continuous random variable numerically using a frequency table and graphically using a frequency histogram and a frequency poly- gon, recognize that the frequency polygon approximates the frequency distribution, and determine, through investigation using tech- nology (e.g., dynamic statistical software, graphing calculator), and compare the effec- tiveness of the frequency polygon as an approximation of the frequency distribution for different sizes of the intervals
2.5 recognize that theoretical probability for a continuous random variable is determined over a range of values (e.g., the probability that the life of a lightbulb is between 90 hours and 115 hours), that the probability that a continuous random variable takes any single value is zero, and that the probabilities of ranges of values form the probability distri- bution associated with the random variable
2.6 recognize that the normal distribution is commonly used to model the frequency and probability distributions of continuous ran- dom variables, describe some properties of the normal distribution (e.g., the curve has a central peak; the curve is symmetric about the mean; the mean and median are equal; approximately 68% of the data values are within one standard deviation of the mean and approximately 95% of the data values are within two standard deviations of the mean), and recognize and describe situations that can be modelled using the normal distribution (e.g., birth weights of males or of females, household incomes in a neighbourhood, baseball batting averages)
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