May 3 2020. Anthony. Probability, Counting Principle, Combinations, and Permutations.
How many ways can a kindergarten teacher line up a group of 8 third-grader students?
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 ways. Arrangements with no restrictions.
There are two spinners. First spinner has numbers 1-12 (2, 5, 8, 11 are shaded while the other numbers are not shaded) while the other spinner has 5 colors: red, blue, green green, red. If each spinner is spun once, A) Find P(11, then red). B) Find P(unshaded, then green). C) Find P(at least 3, then blue. D) Find P(even and shaded).
A) (1/12) (2/5) = 2/60 = 1/30 = 3% B) (8/12) (2/5) = 16/60 = 4/15= 275 C) (10/12) (1/5) = 10/60 = 1/6 =17% D) (2/12) (4/5) = 8/60 = 4/30= 2/15 =13%
7) There are 3 green markers, 6 yellow markers, 4 red markers, and 12 blue markers in a pencil box. A marker is drawn, NOT replaced, then another marker is drawn. Find each probability. A) Find P(red, then blue) B) Find P(yellow, then green) C) Find P(both blue) D) Find P(both yellow)
A) (4/25) (12/24) = 48/600 = 2/25 = 8% B) (6/25) (3/24) = 18/600 = 3/100 = 3% C) (12/25) (12/24) =144/600 = 6/25 24% D) (6/25) (6/24) = 36/600 = 3/50 = 6%
A card is randomly drawn from a standard deck, then a date in the month of June is chosen at random. A) Find P(Ace, then odd). B) Find P(Spade, then a multiple of 4).
A) (4/52) (15/30) = 60/1560 = 1/26 = 4% B) (13/52) (7/30) = 91/1560 = 7/120 = 6%
The spinner is spun once with the options {scalene quadrilateral, square, trapezoid, rhombus, pentagon, triangle, parallelogram, rectangle} available, then a standard die is rolled. Find each probability. A) Find P(quadrilateral, then 6). B) Find P(triangle, then odd). C) Find P(square, then at most 2). D) Find P(parallelogram, then prime)
A) (6/8) (1/6) = 6/48 = 1/8 = 13% B) (1/8) (3/6) = 3/48 = 1/16 = 6% C) (1/8) (2/6) = 2/48 = 1/24 = 4% D) (4/8) (3/6) = 12/48 = 1/4 = 25%
8) A bag contains 16 lottery balls, numbered 1-16. A ball is drawn, NOT replaced, then another ball is drawn. A) Find P(even, then odd) B) Find P(10, then even) C) Find P(less than 13, then 16) D) Find P(both multiples of 5)
A) (8/16) (8/15) = 64/240 = 4/15 =27% B) (1/16) (7/15) = 7/240 = 3% C) (12/16) (1/15) = 12/240 = 1/20 = 5% D) (3/16) (2/15) = 6/240 =1/20 = 5%
11) Patrick is buying a new car. He can choose the body style, color, and engine type. If there are 54 ways he can select a car, with three body styles and two engine choices, how many colors are available?
(3)(x)(2)=54....Multiply 3 and 2. 6x=54......Divide both sides by 6. x=9. There are 9 possible colors to choose from.
12) At the school cafeteria, you can choose one sandwich, one snack, and one drink. The number of drink options is equal to the number of snack options. If there are 63 ways to choose your lunch, with seven different sandwich options, how many drink options do you have?
(7)(x)(x)=63....Multiply. Also note that xx is x^2 (x to the second power) 7y^2=63.....Divide both sides by 7. y^2=9......Opposite of a power of 2 is a root of 2. y=sqrt(9).....The square root of 9 is 3. y=3 There are 3 different drink options available.
How many ways can the letters in the word "COMMISSION" be arranged?
10 total letters. Then there are two M's, two I's, two S's, two O's. 10! / (2! 2! 2! 2!) (10x9x8x7x6x5x4x3x2x1) / (2x2x2x2) 3628800/16 226,800 ways to arrange the letters.
A particular cell phone company offers 6 models of phones: each in 4 different colors and each available with any one of 5 calling plans. How many combinations are possible?
Arrangements: 6 x 4 x 5 = 120 ways.
13) A coin is tossed three times. What is the probability of getting heads just once?
Different outcomes when flipping a coin three times. HHH, HHT, HTT, HTH, TTT, THT, TTH, THH. The only options where Heads occurs once: HTT, THT, TTH. 3/8=38%
16) A standard die is rolled two times. What is the probability that it lands on 1 both times?
Roll one is recorded as 1, 2, 3, 4, 5, 6 as a column. Roll two is recorded as 1, 2, 3, 4, 5, 6 as a row. Sample space is then recorded as {(1,1) (1,2) (1,3), (1,4) (1,5) (1,6) (2,1) (2,2).....(6,6)}. 6rows x 6 columns=36 total possibilities. The only options where 1 and 1 are possible: (1,1). 1/36 = 3%
18) There are two spinners. First spinner has the color options: red, orange, yellow, green, blue. Second spinner has the numbers 1-8 listed once. What is the probability of getting orange, or blue, and a multiple of 3.
Sample space is calculated by counting principle: 5 options for colors times by 8 options for second spinner from 1-8. 5x8= 40 total possibilities. The only options where Orange or Blue occurs AND a multiple of 3. (Orange, 3) (Orange, 6) (Blue, 3) (Blue 6). Four options. 4/40 = 1/10 = 10%
17) There are two spinners. First spinner has the color options: red, orange, yellow, green, blue. Second spinner has the numbers 1-8 listed once. What is the probability of getting yellow and an even number?
Sample space is calculated by counting principle: 5 options for colors times by 8 options for second spinner. 5x8=40. The only options where yellow is spun and an even number occurs (Yellow, 2) (Yellow, 4) (Yellow 6) (Yellow 8). Four options 4/40= 1/10 = 10%
14) A month is chosen at random, then a standard die is rolled. What is the probability of getting February, then a number less than 5?
Sample space of possible events: { (Jan, 1) (Jan, 2), (Jan, 3), (Jan, 4), (Jan, 5), (Jan, 6), (Feb, 1) (Feb, 2), (Feb, 3), (Feb, 4), (Feb, 5), (Feb, 6), (March, 1) (March, 2), (March, 3), (March, 4), (March, 5), (March, 6), .... (Dec 6)} The only options where Feb and the numbers less than 5 occur: (F4, F3, F2, F1) 4/72 = 1/18 = 5%
15) A card from a standard deck is chosen at random, then a coin is tossed. What is the possibility of getting an ace and tails?
There are 52 cards on deck of cards. There are also 2 sides of a coin: 52x2= 104. There are 104 options for the sample space: {(Ace, H) (Ace, T) (2, H) (2, T) (3, H) (3, T)......(King, T)}. The only options where ace and tails are possible: Ace of hearts, Ace of diamonds, ace of spades, ace of club. Then Tails attached. 4/104 = 1/26 = 4%