Binomial distribution questions: formal word problems

Size: px

Start display at page:

Download "Binomial distribution questions: formal word problems"

Dortha Cecilia Barker
5 years ago
Views:

5 Biomial distributio questios: formal word problems For the followig questios, write the iformatio give i a formal way before solvig the problem, somethig like: X = umber of... out of 2, so X B(2, 0.2). The P(X 3) =.... Thirty percet of studets at a large uiversity are kow to be short-sighted. If twety studets are picked at radom, fid the probability that at most two of them are shortsighted. 2. A ubiased coi is tossed 0 times. Fid the probability that there are fewer tha 3 heads tossed. 3. A farmer plats 2 sapligs. O average, 5% of sapligs plated fail to survive their first witer. Fid the probability that more tha oe of his sapligs will die i that first witer. 4. ACME Ic. produces super-tactile widgets for the latest DIY craze, ad they are sold i boxes of 8. Ufortuately, 2% of them are faulty. Fid the probability that a give box has at least oe faulty widget i it. 5. At Covigto High School, 20% of sixth-form mathematics studets retake their C module. There are 5 studets i Mr Smither s class; fid the probability that precisely four of them retake this module. 6. Bria ejoys his pizza. So much so that he eats pizza every ight. He likes both pepperoi ad mushroom, ad each ight he chooses radomly ad idepedetly betwee these two optios. Give that each ight he picks the mushroom optio with probability 0.35, fid the probability that he eats betwee 2 ad 4 (iclusive) mushroom pizzas this week (7 days). 7. The probability of rai o ay give day i Jue i Cambridge is 0.8. Assumig that the weather o each day is idepedet of the weather o other days, fid the probability that it rais o at least 25 days i Jue. 8. There were te gree bottles sittig o the wall. The probability of a gree bottle accidetally fallig is What is the probability that fewer tha 8 of the gree bottles accidetally fall? Aswers: () (2) (3) (4) (5) (6) (7) (8) 0.05

6 Biomial distributio questios: cumulative distributio tables I this sectio, you are give the distributio of X i each questio ad asked to fid a probability usig the cumulative distributio tables.. (a) X B(6, 0.05), fid P(X 3) (b) X B(8, 0.03), fid P(X 2) 2. (a) X B(5, 0.08), fid P(X < 3) (b) X B(9, 0.5), fid P(X < 2) 3. (a) X B(7, 0.2), fid P(X > 3) (b) X B(0, 0.06), fid P(X > ) 4. (a) X B(5, 0.4), fid P(X 2) (b) X B(, 0.3), fid P(X 4) 5. (a) X B(6, 0.3), fid P(X = 2) (b) X B(0, 0.07), fid P(X = ) 6. (a) X B(8, 0.2), fid P( X 4) (b) X B(2, 0.09), fid P(2 X 4) 7. (a) X B(6, 0.8), fid P(X 4) (b) X B(0, 0.6), fid P(X 5) 8. (a) X B(8, 0.7), fid P(X < 5) (b) X B(5, 0.9), fid P(X < 0) (c) X B(5, 0.), fid P(X 5) (d) X B(0, 0.2), fid P(X 3) (c) X B(20, 0.4), fid P(X < 4) (d) X B(2, 0.2), fid P(X < 5) (c) X B(30, 0.), fid P(X > 6) (d) X B(40, 0.25), fid P(X > 8) (c) X B(30, 0.04), fid P(X 3) (d) X B(50, 0.08), fid P(X 5) (c) X B(3, 0.), fid P(X = 3) (d) X B(9, 0.05), fid P(X = 0) (c) X B(5, 0.35), fid P(3 X 6) (d) X B(0, 0.5), fid P( X 3) (c) X B(2, 0.75), fid P(X 7) (d) X B(5, 0.95), fid P(X 3) (c) X B(0, 0.85), fid P(X > 7) (d) X B(6, 0.97), fid P(X 4) Aswers. (a) (b) (c) (d) (a) (b) (c) (d) (a) (b) 0.76 (c) (d) (a) (b) (c) 0.69 (d) (a) (b) (c) (d) (a) (b) (c) (d) (a) 0.90 (b) (c) (d) (a) 0.94 (b) (c) (d) 0.025

7 Biomial Distributio Problems. A study idicates that 4% of America teeagers have tattoos. You radomly sample 30 teeagers. What is the likelihood that exactly 3 will have a tattoo? 2. A XYZ cell phoe is made from 55 compoets. Each compoet has a.002 probability of beig defective. What is the probability that a XYZ cell phoe will ot work perfectly (that at least oe compoet does ot work)? 3. The ABC Compay maufactures toy robots. About toy robot per 00 does ot work. You purchase 35 ABC toy robots. What is the probability that exactly 4 do ot work? 4. The LMB Compay maufactures tires. They claim that oly.007 of LMB tires are defective. What is the probability of fidig 2 defective tires i a radom sample of 50 LMB tires? 5. A HDTV is made from 00 compoets. Each compoet has a.005 probability of beig defective. What is the probability that a HDTV will ot work perfectly? () (a) 20C5 (.08) 5 (.92) 5 =.045 (b) 20C0 (.08) 0 (.92) 20 =.887 (c) 20C20 (.08) 20 (.92) 0 = (ote -22 meas move the decimal 22 places to the left) (2) 30C3 (.04) 3 (.96) 27 =.0863 (3) Probability that it will work (0 defective compoets) 55C0 (.002) 0 (.998) 55 =.896 Probability that it will ot work perfectly is =.04 or 0.4% (4) 35C4 (.0) 4 (.99) 3 = (5) 50C2 (.007) 2 (.993) 48 =.0428 (6) Probability that it will work (0 defective compoets) 00C0 (.005) 0 (.995) 00 =.606 Probability that it will ot work perfectly is =.394 or 39.40%

8 HCP PRECALC SECTION 9.4 NOTES SEQUENCE ordered progressio of umbers EXPLICIT FORM formula for the th term RECURSIVE FORM formula that depeds upo the previous term ARITHMETIC SEQUENCE a sequece with a commo differece a= a a = a + d( ) a = a + d GEOMETRIC SEQUENCE a sequece with a commo ratio g= g g = g r g = g r Example : Fid terms, 2, 3, ad 7 for the followig sequeces. a = + ( ) a = a =! th Example 2: Fid the first three terms for both of the previous sequeces. Also fid the 0 term. a = 000 a =.005 a, > a = a2 = a3 = a 2 + a, > 2 Example 3: For each of the followig sequeces:. Decide if the sequece is arithmetic or geometric 2. Write the explicit ad recursive formulas. 3. Fid the teth term. a. 5,-2,-9,-6,-23, b. 3,2,,,

9 Example 4: Cosider the sequece with t =8, ad t =64. Fid the explicit formula if the sequece is 3 6 arithmetic, or if it is geometric. III. Summatio Notatio A. I summatio otatio, the sum of the terms of the sequece {a, a, a } is deoted 2 Which is read the sum of a from to a k The variable k is called the idex of summatio. A. Example 4: 5 2k = B. Example 5: 2 0 cos( kπ ) = SUM OF A FINITE ARITHMETIC SEQUENCE a + a ak = a+ a a = 2 Example 6: A cocert auditorium has 30 rows of seats. The first row cotais 50 seats. As you move to the rear of the auditorium, each row has two more seats tha the previous oe. How may seats are i the auditorium? SUM OF A FINITE GEOMETRIC SEQUENCE g ( r ) g = g + g g = k 2 r Example 7: Determie the value of the followig partial sum (we say partial, because we are ot addig all terms i the ifiite sequece) ,

10 Cosider the followig geometric sequece: The partial sums are below.,,, a S /2 /2=0.5 2 /4 3/4= /8 7/8= /6 5/6= /32 3/32= /64 63/64= /28 27/28= Note that these umbers seem to approach Takig the limit of the partial sum: INFINITE SERIES sum of a ifiite geometric sequece g gk = g+ g g +... = r k 2 2 ( 0 ) S = lim lim 2 = = = The series CONVERGES if it approaches some umber S. 0< r < The series DIVERGES if it approaches ifiity. r > Example 8: Calculate the sum of the ifiite geometric sequece 2 3 3, 3,3, 3, Example 9: Covert the followig repeatig decimals to fractio form

11 HCP PRECALC Review NAME. Mrs. Joes has give her class a list of the 30 possible problems o the Chapter 2 Test of which she will select 2. If you kow how to correctly aswer 6 of the questios, what is the probability you will be able to aswer the give umber of questios correctly a.) all 2? b.) 7 out of 2? c.) at least 2 of 2 2. Two cas of mixed uts of differet brads are opeed o a table. Brad A cosists of 30% cashews, while brad B cosists of 40% cashews. A ca is chose at radom, ad a ut is chose at radom from the ca. Fid the probability that the ut is? a.) From brad A. c.) a cashew. b.) a brad B cashew. d.) From A, give it is a cashew. 3. The fourth ad eighth terms of a arithmetic sequece are 2 ad 57, respectively. Fid a explicit formula for the th term of this sequece. 4. Write usig summatio otatio assumig the suggested patter cotiues Fid the sum of the followig sequeces: a.) 2, 6, 8,, 39,366 b.) -5, -, 3, 7, Fid the first 4 terms ad the 00 th term of the sequece defied below. a = 3 a = a Evaluate: 9 = State whether the geometric series coverges or diverges. If it coverges, fid its sum k b.) a.) ( ) 2.4k

12 Classwork 9.7: HONORS PRECALCULUS List a set of data for which the iequality holds..) Mea < Mode < Media 2.) Stadard Deviatio < Iterquartile Rage 3.) Rage = Iterquartile Rage Draw a boxplot for which the iequality holds. 4.) Media < Mea 5.) 2 X Iterquartile Rage < Rage 6.) Rage < 2 X Iterquartile Rage Costruct a set of data with a media of 5, mode of 6, ad a mea of 7.

13 HCP PRECALCULUS Chapter 9 REVIEW NAME. Of the 800 math teachers, 60% have a pocket protector, 45% have a calculator ad 200 have both. Draw a Ve diagram represetig this situatio. 2. How may differet 5-letter words ca formed from the letters i the word FUNNY? 3. I the expasio of 23 ( 3x 2), fid the coefficiet of x Fid the probability of drawig the followig from a stadard 52-card playig deck: a.) Pickig a red card. b.) Pickig 2 aces without replacemet. c.) Pickig a kig, give it is a black card. d.) Beig dealt a 5-card had cotaiig 4 aces dice are rolled. Fid the followig probabilities: a.) Rollig exactly 3 fives. b.) Rollig all 0 fives. c.) Rollig at least 2 fives. Fid a explicit formula for the followig: 6.), 3, 6, 0, 5, 7.) 6, 42, 294, 2058,. 8.) Fid the sum of the ifiite geometric series: ) Evaluate (4k 3) k = ) ( k 2k+ ) k =

14 . Prove by mathematical iductio that ( + ) P : ( ) = 2 is true for all positive itegers EXTRA-PRACTICE: State whether the sequece is arithmetic, geometric or either. Fid a explicit formula for the th term of the sequece.. 27, 8, 2, 8, , 4.4,.44,.044, Cosider the sequece defied recursively as: t = 0 t = t + 2 a. Give the first five terms of the sequece: b. Fid a explicit formula for this sequece. 4. Fid the sum of the ifiite geometric series: appropriate formulas(s)) (Use 3 5. Expad ad simplify. 5 i= 2i Evaluate 3k 2 k = 3 Questios 7 8.Express each series usig sigma otatio. 7. cos 2x + cos 4x+ cos 6x+ cos8x+ cos0x

15 9. Express i sigma otatio ad evaluate usig ay appropriate formula from the followig Prove by mathematical iductio (show steps clearly) 0. Prove that for the sequece defied by t = 4, the sum of the first terms ca be foud usig the formula S = (2+ ) for all values of. That is, prove that the followig statemet is true for all : (4i ) = ( 2+ ) i=. Prove by mathematical iductio: (2 + 6) = + 7 i.e., Prove: If a = 2+ 6, the 2 S = + 7

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo