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The Probability of Two Independent Events Occurring Is the Product of the Individual Probabilities

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The Probability of Two Independent Events Occurring Is the Product of the Individual Probabilities Probability: The probability of two independent events occurring is the product of the individual probabilities. (1) A single fair coin is flipped. What is the probability of a head? (2) A fair coin is flipped twice. What is the probability of two consecutive heads? (3) A coin is flipped twice, make a list of all the possible outcomes. (Use H for heads and T for tails.) (4) A fair coin is flipped twice. What is the probability of getting one H and one T (in either order)? (5) A coin is flipped 3 times. Make a list of all the possible outcomes. (6) A fair coin is flipped three times. What is the probability of getting exactly one T ? (7) A fair coin is flipped 3 times. What is the probability of getting two H and one T (in any order). (8) A fair coin is flipped three times. What is the probability of getting at least one H?. Sums of numbers in a progression. Example. Find the sum of all whole numbers greater than 0 and less than 20. Solution: arrange the numbers in pairs so that the sum of each pair is the same: (1 + 19) + (2 + 18) + ::: + (9 + 11) + 10 = 190 There are 9 pairs, each with sum 20. Adding the unpaired number (10) makes the total 190. There is another way to get the same answer. There are 19 numbers, the average is 10, so the total is 19 · 10 = 190 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 " (1) Find the sum of all whole numbers less than 100. (2) Find the sum of all even numbers less than 51. (3) Find the sum of all odd numbers less than 51. (4) Find the sum of all whole numbers less than 51. (5) Find the sum of all odd numbers greater than 50 and less than 100. (6) Find the sum of all multiples of 3 between 20 and 80. 1 Venn Diagrams This is a diagram concerning people watching TV news. Each separate area shows the percentage of people in that category. (1) What do all the percentages add up to? (2) What percentage of people watch channel 6 at 6PM? (3) What percentage of people watch both 6PM and 11 PM news? (4) What percentage of people who watch the 6 PM news on channel 6 also watch the 11 PM news. (5) What percentage of people watch neither the 6 PM news nor the 11 PM news. (6) Of just the people who watch the TV news, what percentage watch channel 6 at 6 PM? (7) What percentage of people don't watch channel 6, 6PM? (8) What percentage of people don't watch TV news at 6PM or 11 PM? (Example) Find the sum of all whole numbers greater than 9 and less than 41. (Solution) Solution: arrange the numbers in pairs so that the sum of each pair is the same: (10 + 40) + (11 + 39) + ::: + (24 + 26) + 25 = 775 There are 15 pairs, each with sum 50. Adding the unpaired number (25) makes the total 750+25 = 775. (9) Find the sum of all whole numbers less than 30. (10) Find the sum of all even numbers less than 30. (11) Find the sum of all odd numbers less than 30. Percentage Problems Review (1) What percent is 5 of 20? (2) What is 25 % of 20? (3) If 5 is 25 % of a number, what is the number? (4) If 12 is 30 % of a number, what is the number? (5) I am going 20 MPH. How fast will I be going if I increase my speed by 25 % ? (6) I am going 20 MPH. How fast will I be going if I decrease my speed by 25 % ? (7) If you grew from 48 inches to 52 inches, what is the percentage increase? 1 (8) If you grew from 4 feet to 4 feet , what is the percentage increase? 3 (9) If the marked price of a sweater is $34.99, and the cost is 20 % off the marked price, how much does it cost? (10) I earn 5 % annually on my savings account. If I had $ 504 at the beginning of the year, how much will it earn? (11) If my savings account earns 5 % per year, computed daily, what is the daily rate? (12) How much do I earn per day on $ 5000 invested at a daily rate of 0.0136986 %? (13) If I earn 7 percent per year and it is reinvested (added to principal) how much would I now have if I started with $ 7000 one year ago? (14) What is 107 % of $ 7000? (15) The tab for dinner is $ 43.25. How much would I spend including 8.7 % sales tax, and 15 % tip on the total (including tax)? 3 (16) The current price of a stock is 10 per share. I paid approximately 25.3 % less than the share 8 price. What price did I pay (rounded to an eighth of a dollar)? Mean, Median, Mode Suppose given the numbers f5; 2; 6; 5; 8g. 5 + 2 + 6 + 5 + 8 26 The Mean is the arithmetic average of these numbers: mean = = = 5:2 5 5 The Median is the middle number. List the numbers in increasing order f2; 5; 5; 6; 8g, and pick the middle one. In this case the median is 5. The Mode is the number that occurs most often. In this example the mode is also 5. 1 Find the mean, median and mode of each of the following sets of numbers. (a) 12, 7, 6, 8, 7 (b) 18, 5, 11, 18, 8, 15, 6, 10, 18, 11, 7. (c) 10, 100, 1000, 853, -30, 746, 123 (2) Find the sum of all whole numbers less than 20. (3) The table give information for Andrew's Basketball team for a nine game season. attempts made 3 point shots 29 11 2 point shots 83 32 free throws 21 10 (3a) How many shots did Andrew average per game? (3b) What percentage of 3 point shots did he make? (3c) What percentage of 2 point shots did he make? (3d) What percentage of free throws did he make (3e) What was his overall shooting percentage? (3f) Free throws score 1 point each. How many points did Andrew score on average? Rolling Dice. (1) How many possible outcomes are there on the roll of one die. (2) How many possible outcomes are there on the roll of a pair of dice. Assume you can distinguish between a 1 on one die, 3 on the second die, and 3 on first die, 1 on the second die. (3) How many outcomes are there on the roll of three dice. (4) Of all possible outcomes of rolling a pair of dice, how many times is the sum equal to 7? What is the probability that the sum is 7? (5) If a pair of fair dice are rolled, what is the probability that the sum is 8? (6) If a pair of fair dice are rolled, what is the probability that the sum is 9? (7) A pair of fair dice are rolled. In the table below list the possible outcomes for the sum, and below each write the probability (each as a fraction with denominator 36). (8) Find the sum of all whole numbers less than 6. (9) What is the sum of the whole numbers less than 6 plus 6? (10) If a pair of fair dice are rolled, what is the probability that the sum is less than 8? (11) If a pair of fair dice are rolled, what is the probability that the sum is greater than 5? (12) A pair of fair dice are rolled, and you see only the sum. What are the possible outcomes? Are the probabilities of each outcome the same. How many times larger is the highest probability (for the sum) than the lowest probability? Marbles, page two You have a bag with 12 marbles. There are 2 Yellow, 4 Blue and 6 Red. The marbles are drawn out of the bag and put in positions 1,2,3. There are 26 outcomes (There would be 27 = 3 × 3 × 3 outcomes,except that there are only 2 Y's, so YYY is excluded.) Now answer the following, in reduced fraction form. (1) What is the probability of YBB? (2) What is the probability of BYB? (3) What is the probability of BBY? (4) What is the probability of 2 B's and a Y in any order? (5) What is the probability of 2 B's and a R in any order? (6) What is the probability of 2, but not 3 B's (7) How many outcomes have 2 but not 3 B's? (8) What is the probability of 2, but not 3 R's (9) What is the probability of 2, but not 3 Y's (10) What is the probability of getting 2 but not 3 of any color? (11) How many outcomes have 2 but not 3 of any one of the colors? (12) How many outcomes have 3 of any one of the colors? (13) How many outcomes have 3 different colors? (14) What is the probability of getting 3 of the same color? (15) What is the probability of getting 2 or more of the same color? Probability and Statistics . (1) Two dice are rolled. What is the probability the sum is 7? (2) What is the mean of 10, 100, 1000, 853, -30 746 123? (3) What is the probability of drawing a 3 or less from a deck of 52 cards (Ace is high).
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