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 {5, 2, 6, 5, 8}.

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 {2, 5, 5, 6, 8}, 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).

(4) What is the probability of getting 3 Heads on 3 flips of a fair coin?

(5) In how many ways can first, second and third places be awarded in a race with 10 competitors.

(6) In a ski club there are 105 members: 61 are expert downhill skiers, 55 are expert cross country skiers, and 23 are expert at both. How many are expert at neither downhill no cross country?

(7) How many different ways can you arrange the letters of the word “Bryant”?

(8) You have some number placards {1, 1, 7, 8, 9}. How many 3 digit numbers can you make?

(9) What is the probability of rolling a sum of 7 on two dice? Playing Cards II

You are dealt 5 cards from a standard 52 card deck (4 suits, 13 cards in each suit). Two cards match if they have the same number (or letter), but are in different suits).

(1) What is the probability of no matches between the 5 cards.

(2) In the case of no matches you had to consider only one case. To calculate the probability of exactly one match, you must consider how many ways one match can happen. If M matches M, and the X’s don’t match M or another X, what are the possible cases? One case is MMXXX.

(3) What is the probability of MMXXX? Are the probabilities of all the cases of (2) the same? Calculate the probability of exactly one match.

(4) Calculate the probability that 3 of the cards match but two don’t. Do this the same way you calculated the 2 cards matching. Determine how many ways it can happen, and the probability for each way.

(5) Calculate the probability that 4 of the cards match.

(6) Calculate the probability of exactly 2 pair. If two pair are represented as AABBC, there are only 15 cases with the same probability that must be counted. There are 30 orders for AABBC, but AABBC and BBAAC are the same, so counting both would double count.

(7) Calculate the probability of a full house AABBB. Since A is part of a pair and B is part of a triple you must count all possible orders.

(8) Summarize your probabilities. All probabilities are fractions with denominator 4165.

numerator

one pair

two pair

three of a kind

full house

4 of a kind Probability and Statistics Concepts Review. Venn diagram of percentages:

(1) A bag contains 10 orange marbles, 5 purple marbles, 7 grey marbles and 3 green marbles. Two marbles are drawn without replacement. What is the probability of getting an orange and a green marble (in either order)?

(2) John and George play a numbers matching game. They each pick a whole number from 1 to 10 and secretly write then down. Then they see if the numbers match. What is the chance of NOT matching on the first trial? What is the chance of not matching on two trials? If someone bets you that John and George will have at least one match in 5 trials, what is your chance of winning the bet?

(3) A loaded die has a 20 % chance of coming up 3 while all other numbers have equal chances. What is the chance of rolling a six?

(4) You are buying skis, boots and bindings. There are 4 choices of skis, 3 choices of boots, and 2 choices of bindings. How many different packages are you considering?

(5) Spinning wheel problem

distance number of from bldg. landings 0 to 5 2 5 to 10 3 (6) A paper airplane was tossed from a building, landing as follows. 10 to 15 7 15 to 20 6 20 to 25 2 25 to 30 3 30 to 35 2 You are asked to predict at what distance from the building it will land on the next toss. What should you say to have the best chance of being correct? Based on the previous tosses, what is the chance you will be correct? Another student is asked if they think you will be short or long, what should they say, and what chance do they have of being correct? Probability and Statistics Review (1) What is the mean, median and mode of {6, 14, 7, 11, 7, 11, 8, 6, 11}?

(2) A drawer contains 2 blue, 2 beige, 2 brown, 2 black and 2 bronze colored socks. If you randomly pick 2 socks, what is the chance you will get a pair (of the same color)?

(3) A bag contains 3 blue, 4 green and 5 red marbles. You pick 3 marbles at random. What is the probability of getting one of each color?

(4) In Mr Jones 5th grade class, the grades for the first term were: Grade Reaading W riting Math Science A 6 5 7 8 B 12 14 11 13 C 8 8 10 6 D 3 2 1 2 (4a) What percentage of the students got an A in reading?

(4b) In what subject did most students get an A or a B?

(4c) If no student got more than 1 D, what percentage of students got at least one D?

(5) You pick a whole number from 300 to 399. What is the probability that a randomly chosen whole number will have exactly one digit that is a 9?

(6) How many ways can 4 students sit around a table? (6a) If the absolute position matters. (6b) If only the relative position and direction matters. (6c) If only the relative position matters.

Probability and Statistics Review Problems.

(1) Find the mean, median and mode of {12, 29, 30, 28, 30, 18, 17, 18, 25, 27, 20, 15, 23, 28, 23, 25, 10, 30, 21}

(2) four cards are drawn from a shuffled 52 card deck. What is the probability of not getting a heart?

(3a) How many possible outcomes are there in 4 tosses of a coin (the order matters).

(3b) If the order doesn’t matter, how many outcomes are there?

(3c) What is the probability of getting at least 2 Heads in 4 tosses of a coin? (4) Diagram showing percentages of native languages in the world.

(4a) What percent speaks both Spanish and English as a native language?

(4a) What percent speaks a native language other than Spanish?

(5) I have 10 playing cards. How many different pairs of 2 cards can I make?

(6) I have 6 digits numbered 1 to 6. How many 6 digit numbers can I make if I only use each digit once?

(1) How many 4 digit numbers can be made if none of the digits are prime? (0146 is not a 4 digit number, but 1046, 1406, and 1460 are all 4 digit numbers.)

(2) What is the chance of never getting a tail on 4 consecutive coin flips?

(3a) A bag has 10 marbles; 5 are green, 3 are blue, 2 are white. If we draw 2 marbles, and the order of drawing doesn’t matter, what are the possible outcomes?

(3b) What is the probability of drawing a blue and a white marble?

(4a) An athletic club has 1107 members. 592 members use the swimming pool weekly, 376 members use the tennis courts weekly, and 289 members don’t use either. How many members use both?

(4b) If you meet a member of the club at a banquet, what is the probability that he swims weekly, but doesn’t play tennis weekly?

(5) What is the probability that rolling two fair dice will result in numbers that add up to 11 or more?

(6) How many rectangles are there in this figure? (Squares are rectangles) Probability and Statistics Test.

(1) A pair of fair dice are rolled. What is the probability that the sum is 3 or higher?

(2) What is the mean of 0.23 0.18 0.92 0.35 0.87 0.39 ?

(3) Two cards are drawn from a regular 52 card deck. What is the probability they are both hearts?

(4) A fair coin is flipped twice. What is the probability both are H or both are T?

(5) There are 1635 students in a high school; 438 are sophomores. There are 184 in band and 65 in Latin. Approximately 32% of the band are sophomores, and approximately 48% of the Latin students are sophomores. What percent of the sophomores take neither band nor Latin?

(6) How many ways can you arrange the letters {A, B, C}

(7) How many ways can you arrange any three letters taken from {A, B, C, D, E, F }?

(8) How many distinct sets of three letter can be made from {A, B, C, D, E, F }? (Order doesn’t matter)

(9) You draw 2 cards from a pack of 52. What is the probability of getting a red 10 and a black 10?

(10) At a basketball game there are 1597 spectators of which 45 are from the press; 974 spectators are for the home team and 424 are for the visiting team. 19 members of the press are rooting for some team. How many of the non-press spectators don’t care who wins? Probability and Statistics Concepts

Statistics - Mean - the sum of all the numbers divided by the number of numbers. - Median - the middle number with the numbers listed in order. - Mode - the number that occurs most often

Percentage diagrams - show relation between groups and subgroups.

Probability - given a set number of equally likely outcomes. 1 Probability of a certain outcome = total number of outcomes

Examples: dice, spinners, well-shuffled cards, coin flips, drawing item from a container. Probability of two given outcomes both happening in two events in specified order is the product of the probability of each one. Probability of one of a number of outcomes in one event is the sum of the probabilities of each one. Determining number of possible of outcomes - try to make diagram or list or identify a pattern.

Consecutive coin tosses outcomes with HT distinct from TH is 2x where x is the number of consec- utive tosses.

Number of different orders of x items all of which are distinct and all of which are included in each outcome is x! = x × (x − 1) × · · · × 2 × 1.

Number of different sets of items with one from each group and all items are different is # of items in group 1 ×# of items in group 2, etc.

Number of ways two items can be selected out of group of n items with order distinguishing = n(n − 1).

n(n − 1) Two out of n with order immaterial = = (n − 1) + (n − 2) + (n − 3) + ··· 2 + 1 2