B. $$ 8 / 10 $$ D. $$ 21 / 30 $$ 8. A card is drawn from a standard deck. Find the probability of drawing a spade or a nine. A. $$ 13 / 13 $$ C. $$ 3 / 13 $$ B. $$ 2 / 13 $$ D. $$ 4 / 13 $$ 9. Which of the following illustrates a mutually exclusive event? A. A probability of getting an odd number or a 3 . B. A probability of getting an even number or a multiple of 3 . C. A probability of getting an odd number or a 6 . D. A probability of getting an even number or a 6 . 10. Which of the following illustrates nonmutually exclusive? A. Rolling an even number or greater than 4. B. Rolling an even number or less than 2. C. Rolling an odd number or greater than 5 . D. Rolling an odd number or divisible by 2 11. What is the probability of drawing a red ace or a heart jack, if a card is drawn at random from a deck of cards? A. $$ 1 / 52 $$ C. $$ 4 / 52 $$ B. $$ 3 / 52 $$ D. $$ 5 / 52 $$ 12. The probabilities of three racers $$ \mathrm{A}, \mathrm{B}, \mathrm{C} $$ of winning the grand prize are $$ 1 / 3,1 / 6 $$, and $$ 1 / 3 $$, respectively. Assuming that only one racer can win the grand prize, find the probability that neither $$ A $$ nor $$ C $$ wine. A. $$ 1 / 4 $$ C. $$ 1 / 5 $$ B. $$ 1 / 3 $$ D. $$ 2 / 3 $$ 13. In a room of 20 students, there are 12 girls and 8 boys. 6 girls and 5 boys are wearing uniform. What is the probability of randomly picking a student who is either wearing uniform or a woman? A. $$ 17 / 20 $$ C. $$ 15 / 20 $$ B. $$ 16 / 20 $$ D. $$ 13 / 20 $$ 14. You roll a fair six-sided die. What is the probability that the die shows an even number or a number greater than three? A. $$ 1 / 3 $$ B. $$ 3 / 3 $$ C. $$ 2 / 3 $$ D. $$ 4 / 3 $$ For items number 15-16. Jissa has a box of pens, of which 3 are red, 8 are blue, and 9 are black. She gets a pen, then returns it to the bax, and then gets a second pen. 15. What kind of event is being illustrated in the given situation? A. Dependent C. Mutually Exclusive B. Independent D. Not Mutually Exclusive 16. What is the probability that the first pen selected is blue and the second one black? A. $$ 9 / 50 $$ B. $$ 8 / 15 $$ C. $$ 7 / 10 $$ D. $$ 1 / 10 $$ For flems \#17-18. A box contains 9 red marbles and 8 white marbles. 17. What is the probability of drawing 2 red marbles and 1 white marble in succession without returning the marbles? A. $$ 9 / 85 $$ B. $$ 10 / 85 $$ C. $$ 11 / 85 $$ D. $$ 12 / 85 $$ 18. What is the probatility of drawing 3 marbles in succession without replacement? A. 20/170 B. $$ 21 / 170 $$ C. $$ 22 / 170 $$ D. 23/170 For items \$19-20. The merchandizer displays 3 flavors of noodles in a grocery store. He had 10 packs of chickon flavor, 20 packs of beof flavor and 12 packs of pork flavor. The merchandizer left and a buyer came. 19. What kind of event is illustrated in the given situation? A. Inclusively Mutual B. Independent C. Dependent D. Exclusively Mutual 20. What is the probability that the buyer buys a pack of chicken flavor, a pack of beef flavor and a pack of pork flavor? A. $$ 10 / 287 $$ B. $$ 100 / 287 $$ C. $$ 12 / 287 $$ D. D. $$ 120 / 287 $$ Prepared by: Checked by: KRISTY P.AY-AD Grade 10 Subject Coordinator Master Teacher II Deped MATATAG Address: F. Torres SL, Davao City Contact No.s (082) 327-8315/ (082) 227-9355 Email Address: davaocity nhsodeped. gov.ph Omfial Facebook: www.fncebook.com/denhsofficial

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Let's solve the problems step by step, extracting the known conditions and calculating the probabilities where necessary.

### Problem 8: Probability of drawing a spade or a nine
A standard deck has 52 cards, with 13 spades and 4 nines (one of which is a spade).

1. **Total spades**: 13
2. **Total nines**: 4
3. **Spade that is also a nine**: 1

Using the formula for the probability of either event occurring:
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
Where:
- $$ P(A) = \frac{13}{52} $$ (probability of drawing a spade)
- $$ P(B) = \frac{4}{52} $$ (probability of drawing a nine)
- $$ P(A \cap B) = \frac{1}{52} $$ (probability of drawing the nine of spades)

Calculating:
$$
P(A \cup B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}
$$

**Answer**: D. $$ \frac{4}{13} $$

### Problem 9: Mutually exclusive events
Mutually exclusive events cannot occur at the same time.

- A. Odd number or a 3: Not mutually exclusive (3 is odd).
- B. Even number or a multiple of 3: Not mutually exclusive (6 is both).
- C. Odd number or a 6: Not mutually exclusive (6 is even).
- D. Even number or a 6: Not mutually exclusive (6 is even).

**Answer**: None of the options illustrate mutually exclusive events.

### Problem 10: Nonmutually exclusive events
Nonmutually exclusive events can occur at the same time.

- A. Even number or greater than 4: Nonmutually exclusive (6 is both).
- B. Even number or less than 2: Nonmutually exclusive (0 is both).
- C. Odd number or greater than 5: Nonmutually exclusive (7 is both).
- D. Odd number or divisible by 2: Mutually exclusive (no number can be both).

**Answer**: A. Rolling an even number or greater than 4.

### Problem 11: Probability of drawing a red ace or a heart jack
There are 2 red aces and 1 heart jack in a deck.

1. **Total red aces**: 2
2. **Total heart jacks**: 1

Using the formula:
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
Where $$ P(A \cap B) = 0 $$ (no overlap).

Calculating:
$$
P(A \cup B) = \frac{2}{52} + \frac{1}{52} = \frac{3}{52}
$$

**Answer**: B. $$ \frac{3}{52} $$

### Problem 12: Probability that neither A nor C wins
The probabilities are:
- $$ P(A) = \frac{1}{3} $$
- $$ P(B) = \frac{1}{6} $$
- $$ P(C) = \frac{1}{3} $$

The probability that neither A nor C wins is:
$$
P(B) = 1 - P(A) - P(C) = 1 - \left(\frac{1}{3} + \frac{1}{3}\right) = 1 - \frac{2}{3} = \frac{1}{3}
$$

**Answer**: B. $$ \frac{1}{3} $$

### Problem 13: Probability of picking a student wearing uniform or a woman
Total students = 20 (12 girls, 8 boys). 
- Girls in uniform: 6
- Boys in uniform: 5

Using the formula:
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
Where:
- $$ P(A) = \frac{12}{20} $$ (girls)
- $$ P(B) = \frac{11}{20} $$ (wearing uniform: 6 girls + 5 boys)
- $$ P(A \cap B) = \frac{6}{20} $$ (girls in uniform)

Calculating:
$$
P(A \cup B) = \frac{12}{20} + \frac{11}{20} - \frac{6}{20} = \frac{17}{20}
$$

**Answer**: A. $$ \frac{17}{20} $$

### Problem 14: Probability of rolling an even number or a number greater than three
Even numbers: 2, 4, 6 (3 outcomes).
Numbers greater than 3: 4, 5, 6 (3 outcomes).
Overlap (even numbers greater than 3): 4, 6 (2 outcomes).

Using the formula:
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
Calculating:
$$
P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{2}{6} = \frac{4}{6} = \frac{2}{3}
$$

**Answer**: C. $$ \frac{2}{3} $$

### Problem 15: Type of event when selecting pens
Since Jissa returns the pen before selecting the second one, the events are independent.

**Answer**: B. Independent

### Problem 16: Probability that the first pen is blue and the second is black
Total pens = 3 + 8 + 9 = 20.

1. Probability of blue first: $$ \frac{8}{20} $$
2. Probability of black second: $$ \frac{9}{20} $$

Calculating:
$$
P(\text{blue first and black second}) = \frac{8}{20} \times \frac{9}{20} = \frac{72}{400} = \frac{9}{50}
$$

**Answer**: A. $$ \frac{9}{50} $$

### Problem 17: Probability of drawing 2 red marbles and 1 white marble without replacement
Total marbles = 9 red + 8 white = 17.

1. Probability of first red: $$ \frac{9}{17} $$
2. Probability of second red: $$ \frac{8}{16} $$
3. Probability of third white: $$ \frac{8}{15} $$

Calculating:
\[
P(\text{2 red, 1 white}) = \frac{9}{17} \times \frac{8}{16} \times \frac{8}{15} = \frac{576}{4080} = \frac{
**Simplified Answers:** 8. **Probability of drawing a spade or a nine:** D. $$ \frac{4}{13} $$ 9. **Mutually exclusive event:** None of the options illustrate mutually exclusive events. 10. **Nonmutually exclusive event:** A. Rolling an even number or greater than 4. 11. **Probability of drawing a red ace or a heart jack:** B. $$ \frac{3}{52} $$ 12. **Probability that neither A nor C wins:** B. $$ \frac{1}{3} $$ 13. **Probability of picking a student wearing uniform or a woman:** A. $$ \frac{17}{20} $$ 14. **Probability of rolling an even number or a number greater than three:** C. $$ \frac{2}{3} $$ 15. **Type of event when selecting pens:** B. Independent 16. **Probability that the first pen is blue and the second is black:** A. $$ \frac{9}{50} $$ 17. **Probability of drawing 2 red marbles and 1 white marble without replacement:** D. $$ \frac{23}{170} $$ 18. **Probability of drawing 3 marbles in succession without replacement:** D. $$ \frac{23}{170} $$ 19. **Type of event when selecting noodles:** C. Dependent 20. **Probability that the buyer buys a pack of chicken, beef, and pork flavors:** D. $$ \frac{120}{287} $$

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