At a mailing center, it was found that on Tuesday, 44 out of 662 envelopes were stamped upside down, on Wednesday, 16 out of 251 envelopes were stamped upside down, and on Thursday, 3 out of 598 envelopes were stamped upside down. Find the probability that an envelope was stamped on Wednesday given that it was stamped right-side up.
SOLUTION
1. Understand Make a table to organize the information. Subtract to find the number of envelopes stamped right-side up.
2. Plan The condition is that the envelope was stamped right-side up. Find the sum of the envelopes stamped right-side up; this will be the denominator.
3. Solve Use a calculator. 235
__
618 + 235 + 595 = 235_
1448 ≈ 0.162 The probability is about 16.2%.4. Check Use estimation to check for reasonableness. From the given information, about 1500 envelopes were stamped and about 250 of those were stamped on Wednesday. The majority of those stamped on Wednesday were stamped right-side up, and _ 250
1500 = 1 _ 6 = 16 2 _ 3 %.
Tues. Wed. Thurs.
Right-side Up 618 235 595
Upside Down 44 16 3
Tues. Wed. Thurs.
Right-side Up 618 235 595
Upside Down 44 16 3
Math Reasoning
Estimate Estimate the probability that an envelope was stamped on Thursday given that it was stamped upside down.
Math Reasoning
Analyze Calculate
P(junior, senior, junior). How does it compare to Example 3?
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Practice
Distributed and IntegratedLesson Practice
Suppose the two spinners shown are each spun once. a. Find the probability that the first
spinner lands on 4 and the product of the spinners is less than 10. b. Find the probability that the first
spinner lands on a number less than 4 and the sum of the numbers is less than 6.
The table shows the results of a survey that asked students from three grade levels to pick their favorite subject from those listed.
c. Find P(Math|12). d. Find P(10th|History).
e. A raffle is being held at a mall. A box
has the names of the 237 men and 318 women who are participating on slips of paper. Gift certificates are being given to the first five names pulled from the box. Once a person wins a certificate, his or her name is not returned to the box. What is the probability that the first three certificates go to men and the last two certificates go to women?
f. At a factory, it was found that during the morning shift, 32 out of the 2666 boxes of nails that were filled had unacceptable weights, and during the evening shift, 11 out of the 1870 boxes of nails that were filled had unacceptable weights. Find the probability that a box was filled in the evening, given that its weight was found acceptable.
4 2 1 3 4 2 1 3 4 2 1 3 4 2 1 3 (Ex 1) (Ex 1) 10th 11th 12th Math 34 56 11 History 45 19 26 Science 52 45 66 English 9 12 58 10th 11th 12th Math 34 56 11 History 45 19 26 Science 52 45 66 English 9 12 58 (Ex 2) (Ex 2) (Ex 3) (Ex 3) (Ex 4) (Ex 4)
1. Solve s2- 22 =-112. Write the solutions in terms of i.
Factor completely.
2. -3pax + pax2 + 2pa 3. 30 + 3x2 - 21x
4. Multi-Step Use x2 - 62x + 961 = 26.
a. Explain why the left side of the equation is a perfect square trinomial. b. Factor x2 - 62x + 961.
c. Solve the equation for x.
(62) (62) (23) (23) (23)(23) (58) (58)
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Lesson 68 487
*5. Error Analysis Find and correct the error a student made in determining the roots and their multiplicity in x3 - 4x2 + 4x = 0.
x3 - 4x2 + 4x = 0
x(x2 - 4x + 4) = 0
x(x - 2)(x - 2) = 0 The only root is 2 and its multiplicity is 2. 6. Analyze What does it mean for an event to have the first number in the odds in
favor ratio greater than the second number?
7. Geometry Write an expanded expression for the volume of the cube shown.
*8. Housing The table shows estimated housing occupancy for three states for the year 2006. Find the probability that a randomly selected
house from one of these states is in Nevada. Then find the probability that a randomly selected house from one of these states is in Nevada given that it is vacant. Round each percent to the nearest tenth.
Simplify. 9. -6 4 - _ 2 3 10.
_
-3 - 9 - _ 32 11. (-8 ) 1 _ 312. Find the arc length of a sector with a radius of 4 in. and central angle θ= _ π 6 .
*13. Probability A bag contains 30 red marbles and 10 white marbles. If three marbles are randomly chosen without replacement, what is the probability of selecting all red or all white marbles?
14. Surveying A surveyor is standing on top of a hill that is 600 meters from a bridge. Her angle of elevation to the top of the bridge is 14° and her angle of depression to the bottom of the bridge is 26°. To the nearest tenth of a meter, what is the height of the bridge?
*15. Generalize Insert = or ≠ to make each statement true. If A and B are independent events, then P(B|A) P(B). If A and B are dependent events, then P(B|A) P(B).
16. Statistics How do probability and statistics compare and contrast? 17. Multiple Choice Which angle does not have a reference angle of 30º?
A -30º B 120º C 570º D -150º (66) (66) (55) (55) (49) (49)
Occupied Houses Vacant Houses
Mississippi 1,075,521 165,918
Nevada 936,828 128,439
New York 7,088,376 819,138
Occupied Houses Vacant Houses
Mississippi 1,075,521 165,918 Nevada 936,828 128,439 New York 7,088,376 819,138 (68) (68) (59) (59) (59)(59) (59)(59) (63) (63) (68) (68) (46) (46) (68) (68) (60) (60) (56) (56) 2b+ 5 cm
SM_A2_NLB_SBK_L068.indd Page 487 6/10/08 1:17:20 PM user-s191
*18. Estimate Use what you know about the unit circle and special angles to estimate Cos-1(0.48) in degrees. Do not use a calculator. Explain your method.
19. Write Describe how to determine cos 150° by using its reference angle. Simplify. 20. 3
√
_
5 2 - 2√
2_
5 21. 3 √ 18 + 2 √ 50 - √ 9822. Calculate the binomial probability P(12, 200, 0.09) to the nearest hundredth of a percent using the formula P(n, N, p)= _ N!
n!(N - n)! p
n(1 - p ) N-n .
*23. Graphing Calculator Describe the feasible region for the set of inequalities x ≥ 0,
x ≤ 4, y ≥ 0, 2y + x ≤ 25.
24. Error Analysis Find and correct the error a student made in subtracting complex numbers. (7 - 4i) - (2 + 3i)
5 - i
*25. Multiple Choice Which describes all the values of sin-1 _ 1 2 ? A
_
5π 6 B π_
6 C_
2π 3 D π_
3 26. Estimate Sketch a graph of the equation y =3
√ 6π2x
_
2π , where y is the radius of a sphere and x is the volume of the sphere. Use the graph to estimate the
volume of a sphere with a radius of 2 cm. Use the graph to estimate the radius of a sphere with a volume of 14 cm3.
27. Volume The volume of a California king size mattress is 12x3- 36x2- 432x - 384
cubic inches. Factor the polynomial given that x - 8 is a factor. Then find the volume given that x = 20.
*28. Plant Nutrition The pH of a soil is given by the function pH =-log [H+], where [H+] represents the concentration of hydrogen ions in moles per liter. Most plants
require soil with a hydrogen ion concentration between 0.0001 and 0.00000001 moles per liter. What pH values correspond to these hydrogen ion concentration values?
*29. Travel A driver leaves a parking lot and heads due south at a rate of 40 miles per hour. An hour later, another driver leaves the same parking lot and heads due west at a rate of 60 miles per hour. Approximately how long will it take for the cars to be 300 miles apart?
*30. Multi-Step Two bags each contain papers labeled with numbers 1 through 5. One paper is selected from each.
a. Explain why selecting a 5 from the first bag and the sum of the numbers is greater than 5 are dependent events.
b. Find the probability of both events.
(67) (67) (63) (63) (44) (44) (44)(44) (Inv 5) (Inv 5) (54) (54) (62) (62) (67) (67) (59) (59) (61) (61) (64) (64) (65) (65) (68) (68)
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Lesson 69 489
Simplifying Complex Expressions
Warm Up
69
L E S S O N
1. Vocabulary A number that can be written in the form a + bi where a and b are real numbers is a .
2. Multiply (3x - 5)(4x + 2). 3. Simplify
_
5√ 5 . 4. Add (3 + 2i) + (6 - 8i).
Complex numbers (a + bi) can be graphed on a complex plane, where the horizontal axis represents the real part, a, and the vertical axis represents the imaginary part, bi.