Page 117 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

B. PROBABILITY DISTRIBUTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of discrete probability distributions, represent them numerically, graphically, and algebraically, determine expected values, and solve related problems from a variety of applications;
2. demonstrate an understanding of continuous probability distributions, make connections to discrete probability distributions, determine standard deviations, describe key features of the normal distribution, and solve related problems from a variety of applications.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 recognize and identify a discrete random vari- able X (i.e., a variable that assumes a unique value for each outcome of a discrete sample space, such as the value x for the outcome of getting x heads in 10 tosses of a coin), gener- ate a probability distribution [i.e., a function that maps each value x of a random variable X to a corresponding probability, P(X = x)] by calculating the probabilities associated with all values of a random variable, with and without technology, and represent a probabil- ity distribution numerically using a table
1.2 calculate the expected value for a given probability distribution [i.e., using
E(X) = ∑ xP(X = x)], interpret the expected value in applications, and make connections between the expected value and the weighted mean of the values of the discrete random variable
Sample problem: Of six cases, three each hold $1, two each hold $1000, and one holds $100 000. Calculate the expected value and interpret its meaning. Make a conjecture about what happens to the expected value if you add $10 000 to each case or if you multi- ply the amount in each case by 10. Verify your conjectures.
1.3 represent a probability distribution graphical- ly using a probability histogram (i.e., a histo- gram on which each rectangle has a base of width 1, centred on the value of the discrete random variable, and a height equal to the probability associated with the value of the random variable), and make connections between the frequency histogram and the probability histogram (e.g., by comparing their shapes)
Sample problem: For the situation involving the rolling of two number cubes and deter- mining the sum, identify the discrete random variable and generate the related probability histogram. Determine the total area of the bars in the histogram and explain your result.
1.4 recognize conditions (e.g., independent trials) that give rise to a random variable that follows a binomial probability distribution, calculate the probability associated with each value of the random variable, represent the distribu- tion numerically using a table and graphically using a probability histogram, and make con- nections to the algebraic representation
P(X=x)= n px(1–p)n–x (x)
Sample problem: A light-bulb manufacturer estimates that 0.5% of the bulbs manufac- tured are defective. Generate and graph the probability distribution for the random vari- able that represents the number of defective bulbs in a set of 4 bulbs.
PROBABILITY DISTRIBUTIONS
1. Understanding Probability Distributions for Discrete Random Variables
115
M a t h e m a t i c s Fo uf nD ca t t i a o Mn as n a g e m e n t
MDM4U