7.5: Conditional Probability
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1 7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) No Gas (G c ) Total a) What is the probability that a person buys a drink? b) What is the probability that a person doesnt buy a drink? c) What is the probability that a person buys gas and a drink? d) What is the probability that a person buys gas but not a drink? e) What is the probability that a person who buys a drink also buys gas? f) What is the probability that a person who doesnt buy a drink buys gas? 1
2 Definition: If E and F are events in an experiment and P (E) 0, then the conditional probability that the event F will occur given that the event E has already occurred is P (F E) = P (E F ). P (E) Definition: The Product Rule is found by rearranging the above formula as follows: P (E F ) = P (E)P (F E). 2
3 Example 2: Lets use a tree diagram to help us understand the product rule: Example 3: At a party, 1 3 of the guest are women. Seventy-five percent of the women wore sandals and 40 % of the men wore sandals. a) What is the probability that a person chosen at random at the party is a man wearing sandals? b) What is the probability that a person chosen at random is wearing sandals? 3
4 Example 4: Consider drawing 3 cards from a standard deck of 52 cards without replacement. a) What is the probability that the 3 cards are hearts? b) What is the probability that the third card drawn is a heart given the first two cards are hearts? 4
5 Independent events Definition: If A and B are independent events, then P (A) = P (A B) and P (B) = P (B A). Thus, two events A and B are independent if and only if P (A B) = P (A) P (B). Example 5: A medical experiment showed the probability that a new medicine was effective was.75, the probability of a certain side effect was.4, and the probability of both occuring was.3. Are the events independent? 5
6 Example 6: Meghan and Natalie go to Freebirds. After ordering their food they get to roll a pair of fair dice for a chance to get their meal for free. Each die has six sides with one of the sides having a backwards F on it. If both of the dice land on the backwards F, you win. What is the probability that at least one of the girls wins? 6
7 7.7 Bayes Theorem Example 7: If we are given information about P (F E), can we find P (E F )? Definition: The above formula is known as Bayes Theorem. 7
8 Example 8: We are to choose a marble from a cup or a bowl. We flip a fair coin to decide whether to choose from the cup or the bowl. The bowl contains 1 red and 2 green marbles. The cup contains 3 red and 2 green marbles. What is the probability that a marble came from the bowl given that it is red? Example 9: A crate contains 7 basketballs and 4 footballs. A bag contains 4 basketballs and 2 footballs. A ball is drawn at random from the crate and put in the bag. A ball is then drawn from the bag. Given that a basketball was chosen from the bag, what is the probability that a football was drawn from the crate? 8
9 Example 10: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card given that the second card is an ace? Example 11:Complete the following tree diagram and use it to answer the following questions: a) Find P (E). b) Find P (A D). c) Find P (B E). d) Are E and B independent events? 9
10 8.1:Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could define the random variable X to represent the number of times we get tails. Example 2: Suppose we roll a die until a 5 is facing up. Then we could define the random variable Y to represent the number of times we rolled the die. Example 3: Suppose a flashlight is left on until the battery runs out. Then we could define the random variable Z to represent the amount of time that passed. 10
11 Types of Random Variables 1. A finite discrete random variable is one which can only take on a limited number of values that can be listed. 2. A infinite discrete random variable is one which can take on an unlimited number of values that can be listed in some sort of sequence. 3. A continuous random variable is one which takes on any of the infinite number of values in some interval of real numbers. (i.e. usually measurements) 11
12 Example 4: Classify each of the following random variables as finite discrete, infinite discrete, or continuous and list/describe the possible values for the random variable. a) A drawer contains 15 red pens, 8 blue pens, and 5 black pens. An experiment consists of drawing pens out of the drawer with replacement until a red pen is drawn. Let X represent the number of pens drawn. b) An experiment consists of randomly selecting 3 pens without replacement from a conveyor belt. Let Y represent the total weight (in ounces) of the selected pens. c) An experiment consists of randomly selecting 3 pens with replacement from a drawer that contains 15 red pens, 8 blue pens, and 5 black pens. Let Z represent the number of red pens drawn. 12
13 Example 5: Referring to Example 1, find the probability distribution of the random variable X. Example 6: An experiment consists of randomly selecting a sample of 3 grapes out of a bowl that contains 20, of which 8 are rotten. Let X represent the number of rotten grapes in the sample. Find the probability distribution of X. 13
14 Example 7: Blue Baker prides itself on having the best chocolate chips cookies in town. To be sure each cookie has the right number of chocolate chips, a few cookies are selected from each batch and the number of chocolate chips in each cookie is counted. This is done for several days and the following results were found: Number of Cookies Number of Chocolate Chips Identify the random variable X in this experiment and find the probability distribution of X. 14
15 We use histograms to represent the probability distributions of random variables. We place the possible values for the random variable X on the horizontal axis. We then center a bar around each x value and let its height be equal to the probability of that x value Example 8: Referring to Example 7, a) Draw the histogram for the random variable X b) Find P (X 12) 15
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