VIDEO 1. A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.

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Amanda Lorin Preston
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1 Part 1: Probability Distributions VIDEO 1 Name: Probability and Binomial Distributions A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled. Recall that the sample space of a random event is the list of all possible outcomes of the event Key Idea: Probability Distribution The sample space of a random variable, with the probabilities associated with all of the values in sample space Example: Key Points to Consider 1. The sum total of all the probabilities in a probability distribution is always equal to 1 2. All probabilities in a probability distribution must be 0 < p < 1 3. Probability distributions of a discrete random variable can be given in a table, graph or probability mass function P(x). Here s one done for us: Example 1) Find k in the following probability distributions: Example 2) Bryana s number of hits in each softball match has the following probability distribution: a) What does x represent? b) Determine and state the value of k c) Determine: i. P(3) ii) P( ) iii) P(1

2 Example 3) The probability distribution of a random variable Y is given by P(Y=y) = cy 3 for y = 1,2,3 Given that c is a constant, find the value of c. Y cy 3 P(Y=y) Part 2. Expected Value of a discrete random variable The expectation of a random variable is the average value you would get if you were to repeatedly measure the variable an infinite number of times The mean or expected value of a random variable is represented by ( and is defined by the following formula: Let s see how it works: Here s one done for us: Example 4) When throwing a standard six-sided dice, let X be the random variable defined by X = the square of the score shown on the dice. Find the exact value of E(X). X P(X=x) Example 5) The random variable V has the probability distribution as shown in the table and E(V) = 6.3. Find the value of k. V k E(V)

If a family consists of 5 children. What is the probability that at most 2 are left handed? What are possible outcomes, how many can be left handed? 0 1 2 3 4 5 Mathematically USE 1. 2 nd VARS 2. 3. TRIALS: 5 p:.

3 Binomial PDFs vs. Binomial CDFs VIDEO 2 Example 6) Compare the following parts: 10% of people are left handed. A. If a family consists of 5 children. What is the probability that exactly 2 are left handed? How many can be left handed? Mathematically USE: 1. 2 nd VARS TRIALS: 5 p:.1 (10% as a decimal) x-value: 2 Type it in! B. If a family consists of 5 children. What is the probability that at most 2 are left handed? What are possible outcomes, how many can be left handed? Mathematically USE 1. 2 nd VARS TRIALS: 5 p:.1 (10% as a decimal) x-value: 2 Type it in! C. If a family consists of 5 children. What is the probability that at least 2 are left handed? What are possible outcomes, how many can be left handed? Mathematically USE We need probability to be:! WHEN given AT LEAST problems, we need to change the inequality sign so that it is! This will also impact what we type in the calculator PREP: 1 P( one less than the x-value ) In calc: 1. 2 nd VARS TRIALS: 5 p:.1 (10% as a decimal) x-value: answer Example 7) X is a random variable such that X ~ B(n,p). Given that the mean of the distribution is 7.8 and p = 0.3 find: a) n b) Hence, the variance of X. c) standard deviation

4 Ticket to Practice: Complete and check in with me to get your hw/practice! 1. An unbiased dice is thrown 10 times. Let x be the number of sixes obtained. Find: a) The expected number of sixes b) the variance of X. 2. The probabilities of Nick scoring home runs in each game during his baseball career are given in the following table. X is the number of home runs per game. a) What is the value of P(2)? b) Determine P(0). c) State the expected value of Nick s homeruns. 3. The probability that Janie takes a bus to work on any morning is 0.4. What is the probability that in a working week of five days she a. Takes the bus only 2 times b. Takes the bus at most 3 times c. Takes the bus at least 2 times.

5 Name: Binomial and Probability Distributions Practice/HW Check key and ALL work! Careful in your calculator! 1. The probability that a telephone line is engaged at a company switchboard is If the switchboard has 10 lines, find the probability that a) Exactly one half of the lines are engaged b) At most three lines are engaged c) At least 4 lines are engaged 2. In a large university the probability that a student is left handed is A sample of 150 students is randomly selected from the university. Let k be the expected number of left handed students in this sample. a. Find K. b. i) Hence, find the probability that exactly k students are left handed ii) Hence find the probability that fewer than k students are left handed c. Determine the standard deviation of the sample.

6 3. A factory has two machines, A and B. The number of breakdowns of each machine is independent from day to day. Let A be the number of breakdowns of Machine A on any given day. The probability distribution for A can be modeled by the following table. Let B be the number of breakdowns of machine B on any given day. The probability distribution for B can be modeled by the following table. On Tuesday, the factory uses both Machine A and Machine B. The variables A and B are independent. a) Find k. b) i) A day is chosen at random. Write down the probability that Machine A has no breakdowns ii)five days are chosen at random find the probability that Machine A has no breakdowns on exactly four of these days. c) Find E(B) d) Find the probability that there are exactly two breakdowns on Tuesday.

7 4. The following table shows the probability distribution of a discrete random variable X. a. Find the value of k b. Find E(x) 5. A. The probability of obtaining tails when a biased coin is tossed is The coin is tossed ten times. Find the probability of obtaining at least four tails. B. The Probability of obtaining tails when a biased coin is tossed is The coin is tossed ten times. Find the probability of obtaining the fourth tail on the tenth toss.

8 6. A multiple choice test consists of 10 questions. Each question has five answer choices. Only one of the answers is correct. For each question, Jose randomly chooses one of the five answer. a. Find the expected number of questions Jose answers correctly. b. Find the probability that Jose answers exactly three questions correctly. c. Find the probability that Jose answers more than three questions correctly. 7. The random variable X has the following probability distribution with P(X>1) = 0.5. a. Find the value of r. b. Given that E(X) = 1.4, find the value of p and of q.

9 8. A random variable X has its probability distribution given by P(X =x) = k(x+3) where x is 0,1,2, or 3. SIMILAR TO #3 in video 1! Set up a table if it helps! a) Show that k =. b) Find the exact value of E(X). 9. The probability that Nicole goes to bed at 10:30 on a given day is 0.4. Calculate the probability that on five consecutive days she goes to bed at 10:30 on at most three days. 10. Paula goes to work three days a week. On any day, the probability that she goes on a red bus is. a. Write down the expected number of times that Paula goes to work on a red bus in one week. b. In one week, find the probability that she goes to work on a red bus on exactly two days. c. In one week, find the probability that she goes to work on a red bus on at least one day.

Math 14 Lecture Notes Ch. 4.3

Math 14 Lecture Notes Ch. 4.3 4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or