EXERCISE TOPIK 4 : PROBABILITY PROBABILITY OF AN EVENT 1. Ten balls are numbered 1 to 10 and placed in a box. Balls number 1 to 4 are red, while balls number 5 to 10 are blue, A ball is randomly drawn from the box. What is the probability that the ball drawn is :- a. Blue in color? b. Red in colour? c. With the number greater than 6 ? 2. A letter is randomly drawn from a set of 6 letters $$ \{A, B, C, D, E, F\} $$. Find the probability of each of the following events. a. The letter is an A b. The letter is vowel c. The letter is not A, B or D 3. Two fair dices are rolled. If X is a random variable which represents the sum of the two numbers of the dice, find the probability that a. $$ X=3 $$ b. $$ X=10 $$ c. $$ X $$ is less than 4 d. X is not 12 e. X is 2 or 12 4. A marble is drawn randomly from a bag containing 13 white marbles, 7 blue marbles and 10 yellow marbles. Find the probability that the marble drawn is a. Blue b. Yellow c. Blue or yellow d. Not white e. White or blue f. Green 5. Three fair coins are tossed. Find the probability of each of the following events : a. Three heads are obtained b. Exactly one head is obtained c. Exactly two heads are obtained d. No heads are obtained 6. A pair of fair dice are rolled. What is the probability of getting a sum of a. 2 b. 12 c. 2 or 12

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Let's solve each of the problems step by step, extracting the known conditions and calculating the probabilities accordingly.

### Problem 1: Probability of Drawing Balls

1. **Total Balls**: 10 (numbered 1 to 10)
   - Red Balls: 4 (numbered 1 to 4)
   - Blue Balls: 6 (numbered 5 to 10)

**a. Probability of drawing a blue ball:**
$$
P(\text{Blue}) = \frac{\text{Number of Blue Balls}}{\text{Total Balls}} = \frac{6}{10} = 0.6
$$

**b. Probability of drawing a red ball:**
$$
P(\text{Red}) = \frac{\text{Number of Red Balls}}{\text{Total Balls}} = \frac{4}{10} = 0.4
$$

**c. Probability of drawing a ball with a number greater than 6:**
- Balls greater than 6: 7, 8, 9, 10 (4 balls)
$$
P(\text{Number} > 6) = \frac{4}{10} = 0.4
$$

### Problem 2: Probability of Drawing Letters

1. **Total Letters**: 6 (A, B, C, D, E, F)

**a. Probability of drawing an A:**
$$
P(A) = \frac{1}{6}
$$

**b. Probability of drawing a vowel (A, E):**
- Vowels: 2 (A, E)
$$
P(\text{Vowel}) = \frac{2}{6} = \frac{1}{3}
$$

**c. Probability of drawing a letter that is not A, B, or D:**
- Letters not A, B, or D: C, E, F (3 letters)
$$
P(\text{Not A, B, or D}) = \frac{3}{6} = \frac{1}{2}
$$

### Problem 3: Probability with Two Dice

1. **Total Outcomes**: 36 (6 sides on each die)

**a. Probability that $$ X = 3 $$:**
- Combinations: (1,2), (2,1) → 2 outcomes
$$
P(X=3) = \frac{2}{36} = \frac{1}{18}
$$

**b. Probability that $$ X = 10 $$:**
- Combinations: (4,6), (5,5), (6,4) → 3 outcomes
$$
P(X=10) = \frac{3}{36} = \frac{1}{12}
$$

**c. Probability that $$ X < 4 $$:**
- Combinations: (1,1), (1,2), (2,1) → 3 outcomes
$$
P(X < 4) = \frac{3}{36} = \frac{1}{12}
$$

**d. Probability that $$ X $$ is not 12:**
- Only one combination for 12: (6,6) → 35 outcomes for not 12
$$
P(X \neq 12) = \frac{35}{36}
$$

**e. Probability that $$ X = 2 $$ or $$ X = 12 $$:**
- Combinations for 2: (1,1) → 1 outcome
- Combinations for 12: (6,6) → 1 outcome
$$
P(X=2 \text{ or } X=12) = \frac{1 + 1}{36} = \frac{2}{36} = \frac{1}{18}
$$

### Problem 4: Probability of Drawing Marbles

1. **Total Marbles**: 30 (13 white, 7 blue, 10 yellow)

**a. Probability of drawing a blue marble:**
$$
P(\text{Blue}) = \frac{7}{30}
$$

**b. Probability of drawing a yellow marble:**
$$
P(\text{Yellow}) = \frac{10}{30} = \frac{1}{3}
$$

**c. Probability of drawing a blue or yellow marble:**
$$
P(\text{Blue or Yellow}) = P(\text{Blue}) + P(\text{Yellow}) = \frac{7}{30} + \frac{10}{30} = \frac{17}{30}
$$

**d. Probability of not drawing a white marble:**
$$
P(\text{Not White}) = \frac{7 + 10}{30} = \frac{17}{30}
$$

**e. Probability of drawing a white or blue marble:**
$$
P(\text{White or Blue}) = P(\text{White}) + P(\text{Blue}) = \frac{13}{30} + \frac{7}{30} = \frac{20}{30} = \frac{2}{3}
$$

**f. Probability of drawing a green marble:**
- No green marbles in the bag.
$$
P(\text{Green}) = 0
$$

### Problem 5: Probability with Three Coins

1. **Total Outcomes**: 8 (2^3)

**a. Probability of getting three heads:**
- Only one outcome: HHH
$$
P(\text{Three Heads}) = \frac{1}{8}
$$

**b. Probability of getting exactly one head:**
- Outcomes: HHT, HTH, THH → 3 outcomes
$$
P(\text{Exactly One Head}) = \frac{3}{8}
$$

**c. Probability of getting exactly two heads:**
- Outcomes: HHT, HTH, THH → 3 outcomes
$$
P(\text{Exactly Two Heads}) = \frac{3}{8}
$$

**d. Probability of getting no heads:**
- Only one outcome: TTT
$$
P(\text{No Heads}) = \frac{1}{8}
$$

### Problem 6: Probability with Two Dice (Sum)

1. **Total Outcomes**: 36 (6 sides on each die)

**a. Probability of getting a sum of 2:**
- Only one combination: (1,1)
$$
P(\text{Sum} = 2) = \frac{1}{36}
$$

**b. Probability of getting a sum of 12:**
- Only one combination: (6,6)
$$
P(\text{Sum} = 12) = \frac{1}{36}
$$

**c. Probability of getting a sum of 2 or 12:**
\[
P(\text{Sum} = 2 \text{ or } 12) = P(\text{Sum} = 2) + P(\text{Sum} = 12) = \frac{1}{36} + \frac{1}{36} = \frac{2}{36} = \frac{1}{
**Problem 1: Probability of Drawing Balls** - **a. Probability of drawing a blue ball:** 60% - **b. Probability of drawing a red ball:** 40% - **c. Probability of drawing a ball with a number greater than 6:** 40% **Problem 2: Probability of Drawing Letters** - **a. Probability of drawing an A:** 16.67% - **b. Probability of drawing a vowel (A, E):** 33.33% - **c. Probability of drawing a letter that is not A, B, or D:** 50% **Problem 3: Probability with Two Dice** - **a. Probability that the sum is 3:** 2.78% - **b. Probability that the sum is 10:** 8.33% - **c. Probability that the sum is less than 4:** 8.33% - **d. Probability that the sum is not 12:** 97.22% - **e. Probability that the sum is 2 or 12:** 2.78% **Problem 4: Probability of Drawing Marbles** - **a. Probability of drawing a blue marble:** 23.33% - **b. Probability of drawing a yellow marble:** 33.33% - **c. Probability of drawing a blue or yellow marble:** 56.67% - **d. Probability of not drawing a white marble:** 56.67% - **e. Probability of drawing a white or blue marble:** 66.67% - **f. Probability of drawing a green marble:** 0% **Problem 5: Probability with Three Coins** - **a. Probability of getting three heads:** 12.5% - **b. Probability of getting exactly one head:** 37.5% - **c. Probability of getting exactly two heads:** 37.5% - **d. Probability of getting no heads:** 12.5% **Problem 6: Probability with Two Dice (Sum)** - **a. Probability of getting a sum of 2:** 2.78% - **b. Probability of getting a sum of 12:** 2.78% - **c. Probability of getting a sum of 2 or 12:** 2.78%

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