May 14, 009 Independent events page 1 Independent events In the last lesson we were finding the probability that a 1st event happens and a nd event happens by multiplying two probabilities For all the problems in that packet, the probability of the nd event was dependent on the outcome of the first event That is, we had to know the 1st event s outcome before we could find the probability for the nd event But suppose the two events involved in a problem are independent (meaning that they are unrelated to each other, in the sense that knowing the outcome of the 1st event doesn t change the probabilities for the nd event) Then finding the answer to an and probability problem is much simpler: you can just find two probabilities separately, then multiply Here s the rule (with P standing for Probability of ), followed by some examples Rule for independent events: If A and B are independent events, P(A and B) P(A) P(B) Example: Suppose that you flip a coin and roll a 6-sided die What is the probability that the coin flip is tails and the die roll is a 5? Solution: The coin flip and the die roll are independent The probability that the coin flip is tails is 1 The probability that the die roll is a 5 is 61 1 1 Answer: 1 6 1 Example: In a refrigerator, there are: 3 regular sodas, 7 diet sodas, 7 apples, and 5 pears Someone randomly chooses a soda and a fruit What is the probability that a diet soda and a pear were taken? Solution: The soda choice and the fruit choice are independent The probability that the soda is a diet soda is 107 The probability that the fruit is a pear is 15 7 35 Answer: 5 10 1 10 or 4 7 Example: A 6-sided die is rolled twice What is the probability of rolling a 3 on the first roll and a 5 or a 6 on the second roll? Solution: The two dice rolls are independent The probability that the first roll is a 3 is 61 The probability that the second roll is a 5 or a 6 is 6 1 Answer: 6 6 36 or 181
May 14, 009 Independent events page You try it 1 One jar has 5 red marbles and 3 yellow marbles Another jar has 4 green marbles and 6 blue marbles Suppose that one marble is randomly drawn from each jar a What is the probability of getting a red marble and a green marble? b What is the probability of getting a red marble and a blue marble? 3 18 c Write a question about the marbles whose answer would be 6 8 10 80 A drawer contains: 5 green tennis balls, pink tennis balls, 3 orange racquetballs, and 6 red racquetballs Suppose that a tennis ball and a racquetball are randomly chosen a What is the probability of getting a pink tennis ball and an orange racquetball? b What is the probability of getting a green tennis ball and a red racquetball? 1 c Write a question about the balls whose answer would be 63
May 14, 009 Independent events page 3 3 The table shows how many boys and how many girls our school has in 9th and 10th grades Suppose that a 9th grader and a 10th grader are chosen to win an award a What is the probability that the 9th grader chosen is a boy? boys girls 9th 40 57 10th 48 45 b What is the probability that the 9th grader chosen is a girl? c What is the probability that the 10th grader chosen is a boy? d What is the probability that the 10th grader chosen is a girl? e What is the probability that a 9th grade girl and 10th grade boy were chosen? f What is the probability that a 9th grade boy and 10th grade girl were chosen? g What is the probability that girls are chosen from both 9th grade and 10th grade? h What is the probability that boys are chosen from both 9th grade and 10th grade?
May 14, 009 Independent events page 4 4 Suppose you roll two 6-sided dice One die is red and the other die is green a What is the probability of rolling a 3 on the red die and a 5 on the green die? b What is the probability of rolling a 3 on the green die and a 5 on the red die? c What is the probability of rolling a 3 and a 5 on the two dice? Hint: Combine the answers from parts a and b d What is the probability of rolling a 4 and a 4 on the two dice? e Parts c and d should have had different answers (If you got the same answer for both, go back and reconsider) Explain what was different about the two problems that made the answers come out different The method of multiplying probabilities extends to situations where there are more two events For example, if you are asked the probability that three events all happen, multiply the three probabilities 5 Suppose a refrigerator contains: 3 regular sodas, 7 diet sodas, 7 apples, 5 pears, turkey sandwiches, and 1 veggie sandwich Suppose someone randomly chooses a soda, a fruit, and a sandwich a What is the probability of choosing a regular soda, an apple, and a veggie sandwich? 3 4 b Write a problem whose answer would be 7 10 1 3 360
May 14, 009 Independent events page 5 6 Suppose you roll three 6-sided dice a What is the probability of rolling a 4 on all three dice? b Suppose the dice are colored red, green, and blue What is the probability of rolling a on the red die, a 5 on the green die, and a 6 on the blue die? c What is the probability of rolling a, a 5, and a 6 where any number could be on any of the dice? Hint: First write down the several different ways this could happen (one of the ways was shown in part b; the others involve switching around which number goes with which color) Find the probability of each of these ways, then add d What is the probability of rolling a 1, a, and a?
May 14, 009 Independent events page 6 7 In these problems about coin flipping, H stands for heads and T stands for tails a Suppose you flip a coin 3 times What is the probability that it comes up H all three times? b Suppose you flip a coin 3 times What is the probability that the sequence of flips is HTH? c Suppose you flip a coin 4 times What is the probability that it comes up T all four times? d Suppose you flip a coin 4 times What is the probability that the sequence of flips is TTHH? e Suppose you flip a coin 5 times Write a question about the flips whose answer would be 1 1 1 1 1 1 3 f Fill in this table showing the probabilities that when a coin is flipped a certain number of times, all of the flips will be H number of flips 1 3 4 5 6 7 8 probability that all the flips are H s g Suppose that a coin is flipped n times What is the probability that the coin flip will be H every time? h How many times would you need to flip the coin, for the probability of getting all H s to be less than 0001?
May 14, 009 Independent events page 7 The remaining problems have some situations where the events are independent (like today s assignment) and some situations where the events are dependent (like yesterday s assignment; remember that tree diagrams can be helpful in answering that kind of problem) 8 Tom has 9 pens in his backpack: 6 blue and 3 red Here are two slightly different questions about the pens (the only difference is highlighted in bold) a Tom randomly takes a pen from his backpack to take notes in English He puts the pen away at the end of class Next period in Social Studies, again he randomly takes a pen from his backpack What is the probability that Tom used a red pen in English and a blue pen in Social Studies? b Tom randomly takes a pen from his backpack to take notes in English He forgets to put that pen away, leaving it on his English desk Next period in Social Studies, again he randomly takes a pen from his backpack What is the probability that Tom used a red pen in English and a blue pen in Social Studies?
May 14, 009 Independent events page 8 9 A child s toy box contains 6 rectangle blocks and 4 triangle blocks a Suppose that a block is taken from the toy box, and not put back Then, another block is taken from the toy box What is the probability that a triangle block was taken the first time and a rectangle block was taken the second time? b Suppose that a block is taken from the toy box, and then returned Then, a block is taken from the toy box again What is the probability that a triangle block was taken the first time and a rectangle block was taken the second time? c Suppose that a block is taken from the toy box, and not put back Then, another block is taken from the toy box What is the probability that both blocks taken are rectangles? d Suppose that a block is taken from the toy box, and then returned Then, a block is taken from the toy box again What is the probability that both blocks taken are rectangles?