Events thatcannot happen at the same timeare calledmutually exclusiveevents. For example, a number cannot be both even and odd or you cannot have picked a single card from a deck of cards that is both a ten and a jack.
Mutually inclusiveevents, however, can occur at the same time. For example a number can be both less than 5 and even or you can pick a card from a deck of cards that can be a club and a ten.
When finding the probability of events occurring at the same time, there is a concept known as the “double counting” feature. It happens when the intersection is counted twice.
In mutually exclusive events,P(A∩B) =φ, because they cannot happen at the same time.
To find the probability of either mutually exclusive eventsAorBoccurring, use the following formula.
To find the probability of one or the other mutually exclusive or inclusive events, add the individual probabilities and subtract the probability they occur at the same time.
P(A or B) =P(A) +P(B)−P(A∩B) Example: Two cards are drawn from a deck of cards. Let:
A: 1stcard is a club
1.8. Probability of Compound Events www.ck12.org
C: 2nd card is a heart
Find the following probabilities: (a)P(AorB) (b)P(BorA) (c)P(AandC) Solution: (a)P(A or B) =1352+524 − 1 52 P(A or B) = 1652 P(A or B) = 134 (b)P(B or A) =524 +1352− 1 52 P(B or A) = 1652 P(B or A) = 134 (c)P(A and C) =1352×1352 P(A and C) = 2704169 P(A and C) = 161
Practice Set
1. Defineindependent events.
Are the following events independent or dependent? 2. Rolling a die and spinning a spinner
3. Choosing a book from the shelf then choosing another book without replacing the first 4. Tossing a coin six times then tossing it again
5. Choosing a card from a deck, replacing it, and choosing another card
6. If a die is tossed twice, what is the probability of rolling a 4 followed by a 5? 7. Definemutually exclusive.
Are these events mutually exclusive or mutually inclusive? 8. Rolling an even and an odd number on one die.
9. Rolling an even number and a multiple of three on one die. 10. Randomly drawing one card and the result is a jack and a heart. 11. Randomly drawing one card and the result is black and a diamond. 12. Choosing an orange and a fruit from the basket.
13. Choosing a vowel and a consonant from a Scrabble bag.
14. Two cards are drawn from a deck of cards. Determine the probability of each of the following events: a. P(heart or club)
b. P(heart and club) c. P(red or heart) d. P(jack or heart) e. P(red or ten)
15. A box contains 5 purple and 8 yellow marbles. What is the probability of successfully drawing, in order, a purple marble and then a yellow marble?{Hint: In order means they are not replaced.}
16. A bag contains 4 yellow, 5 red, and 6 blue marbles. What is the probability of drawing, in order, 2 red, 1 blue, and 2 yellow marbles?
17. A card is chosen at random. What is the probability that the card is black and is a 7?
Mixed Review
18. A circle is inscribed within a square, meaning the circle’s diameter is equal to the square’s side length. The length of the square is 16 centimeters. Suppose you randomly threw a dart at the figure. What is the probability the dart will land in the square, but not in the circle?
19. Why is 7−14x4+7xy5−1x−1=8x2y3not considered a polynomial? 20. Factor 72b5m3w9−6(bm)2w6.
21. Simplify 25−73a3b7+35a3b7−23.
22. Bleach breaks down cotton at a rate of 0.125% with each application. A shirt is 100% cotton. a. Write the equation to represent the percentage of cotton remaining afterwwashes. b. What percentage remains after 11 washes?
c. After how many washes will 75% be remaining? 23. Evaluate (1009−÷24××32+−2492)2.
1.9. Chapter 9 Review www.ck12.org
1.9
Chapter 9 Review
Define the following words: 1. Polynomial 2. Monomial 3. Trinomial 4. Binomial 5. Coefficient 6. Independent events 7. Factors 8. Factoring
9. Greatest common factor 10. Constant
11. Mutually exclusive 12. Dependent events
Identify the coefficients, constants, and the polynomial degrees in each of the following polynomials. 13. x5−3x3+4x2−5x+7
14. x4−3x3y2+8x−12
Rewrite the following in standard form. 15. −4b+4+b2
16. 3x2+5x4−2x+9
Add or subtract the following polynomials and simplify. 17. Addx2−2xy+y2and 2y2−4x2and 10xy+y3. 18. Subtractx3−3x2+8x+12 from 4x2+5x−9. 19. Add 2x3+3x2y+2yandx3−2x2y+3y
Multiply and simplify the following polynomials. 20. (−3y4)(2y2) 21. −7a2bc3(5a2−3b2−9c2) 22. −7y(4y2−2y+1) 23. (3x2+2x−5)(2x−3) 24. (x2−9)(4x4+5x2−2) 25. (2x3+7)(2x3−7)
Square the binomials and simplify. 26. (x2+4)2
27. (5x−2y)2
Solve the following polynomial equations. 29. 4x(x+6)(4x−9) =0
30. x(5x−4) =0
Factor out the greatest common factor of each expression 31. −12n+28n+4 32. 45x10+45x7+18x4 33. −16y5−8y5x2+40y6x3 34. 15u4−10u6−10u3v 35. −6a9+20a4b+10a3 36. 12x+27y2−27x6
Factor the difference of squares. 37. x2−100
38. x2−1 39. 16x2−25 40. 4x2−81
Factor the following expressions completely. 41. 5n2+25n
42. 7r2+37r+36 43. 4v2+36v
44. 336xy−288x2+294y−252x
45. 10xy−25x+8y−20 Complete the following problems.
46. One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the right triangle.
47. A rectangle has sides ofx+5 andx−3. What value ofxgives an area of 48?
48. Are these two events mutually exclusive, mutually inclusive, or neither? “Choosing the sports section from a newspaper” and “choosing the times list for the movie theatre”
49. You spin a spinner with seven equal sections numbered one through sevenandroll a six-sided cube. What is the probability that you roll a five on both the cube and the spinner?
50. You spin a spinner with seven equal sections numbered one through sevenandroll a six-sided cube. Are these eventually mutually exclusive?
51. You spin a spinner with seven equal sections numbered one through sevenandroll a six-sided cube. Are these events independent?
52. You spin a spinner with seven equal sections numbered one through sevenandroll a six-sided cube. What is the probability you spin a 3, 4, or 5 on the spinnerorroll a 2 on the cube?
1.10. Chapter 9 Test www.ck12.org
1.10
Chapter 9 Test
Simplify the following expressions. 1. (4x2+5x+1)−(2x2−x−3) 2. (2x+5)−(x2+3x−4) 3. (b+4c) + (6b+2c+3d) 4. (5x2+3x+3) + (3x2−6x+4) 5. (3x+4)(x−5) 6. (9x2+2)(x−3) 7. (4x+3)(8x2+2x+7) Factor the following expressions.
8. 27x2−18x+3 9. 9n2−100 10. 648x2−32 11. 81p2−90p+25 12. 6x2−35x+49
Solve the following problems.
13. A rectangle has sides ofx+7 andx−5. What value ofxgives an area of 63?
14. The product of two positive numbers is 50. Find the two numbers if one of the numbers is 6 more than the other.
15. Give an example of two independent events. Determine the probability of each event. Use it to find: a. P(A∪B)
b. P(A∩B)
16. The probability it will rain on any given day in Seattle is 45%. Find the probability that: a. It will rain three days in a row.
b. It will rain one day, not the next, and rain again on the third day.