17. A pair of dice are rolled. One die is black and one is white. What is the probability that: a. You roll a 6 on the black and a 5 on the white? b. Both die are less than 4 ? c. The black die is at least 5 and the white die is at most 4 ? d. You roll two odd numbers? e. You roll an even on the black or an odd on the white? 18. A drawer contains 5 black ties and 3 red ties. Suppose you reach in a drawer and draw two ties at random. What is the probability of: a. Drawing two black ties? b. Drawing one of each color? c. Drawing two red ties? 19. A box contains 6 good bulbs and 2 defective bulbs. You reach in the box and draw two bulbs at random. What is the probability of: a. Drawing two defective bulbs? b. Drawing two good bulbs? c. Drawing one of each?

Below are the step‐by‐step solutions for each problem. ──────────────────────────── 17. Two dice are rolled: one die is black and one is white. There are 6 outcomes for each die, for a total of 36 equally likely outcomes. a. Roll a 6 on the black die and a 5 on the white.   • Only one outcome: (black = 6, white = 5)   • Probability = 1⁄36 b. Both dice show a number less than 4.   • Numbers less than 4 are 1, 2, or 3.   • Black die: 3 outcomes; White die: 3 outcomes → 3 × 3 = 9 outcomes.   • Probability = 9⁄36 = 1⁄4 c. The black die is at least 5 and the white die is at most 4.   • Black die “at least 5” means 5 or 6 → 2 outcomes.   • White die “at most 4” means 1, 2, 3, or 4 → 4 outcomes.   • Total outcomes = 2 × 4 = 8   • Probability = 8⁄36 = 2⁄9 d. You roll two odd numbers.   • Odd numbers on a die: 1, 3, 5 → 3 outcomes per die.   • Total outcomes = 3 × 3 = 9   • Probability = 9⁄36 = 1⁄4 e. You roll an even on the black die or an odd on the white die.   • It’s easier to use the complement: the only time the event fails is if the black die is odd AND the white die is even.   • Black odd: 3 outcomes (1, 3, 5) → probability = 3⁄6 = 1⁄2.   • White even: 3 outcomes (2, 4, 6) → probability = 3⁄6 = 1⁄2.   • Thus the probability both occur = (1⁄2) × (1⁄2) = 1⁄4.   • So, probability of the given event = 1 – 1⁄4 = 3⁄4.   • (Alternatively, using the inclusion‐exclusion principle gives the same result.) ──────────────────────────── 18. A drawer contains 5 black ties and 3 red ties (total = 8 ties). Two ties are drawn without replacement. a. Drawing two black ties:   • Ways to choose 2 black ties = C(5, 2) = 10.   • Total ways to choose any 2 ties = C(8, 2) = 28.   • Probability = 10⁄28 = 5⁄14 b. Drawing one tie of each color:   • Ways: choose 1 black (5 ways) and 1 red (3 ways) = 5 × 3 = 15.   • Probability = 15⁄28 c. Drawing two red ties:   • Ways to choose 2 red ties = C(3, 2) = 3.   • Probability = 3⁄28 ──────────────────────────── 19. A box contains 6 good bulbs and 2 defective bulbs (total = 8 bulbs). Two bulbs are drawn at random. a. Drawing two defective bulbs:   • Ways = C(2, 2) = 1.   • Total ways = C(8, 2) = 28.   • Probability = 1⁄28 b. Drawing two good bulbs:   • Ways = C(6, 2) = 15.   • Probability = 15⁄28 c. Drawing one good bulb and one defective bulb:   • Ways: choose 1 good (6 ways) and 1 defective (2 ways) = 6 × 2 = 12.   • Probability = 12⁄28 = 3⁄7 ──────────────────────────── Final Answers: 17a. 1⁄36 17b. 1⁄4 17c. 2⁄9 17d. 1⁄4 17e. 3⁄4 18a. 5⁄14 18b. 15⁄28 18c. 3⁄28 19a. 1⁄28 19b. 15⁄28 19c. 3⁄7 Any answer written in an equivalent form is correct.