RE - $$ 103 I^{\prime} \mathrm{KoP} $$ एउटा बासकमा 6 ओटा सेता र 10 ओटा काला उस्तै र उन्रै बलहरू छन् । दुई ओटा बलहरू एकपछि अर्को गरी पुन: राखेर भिकिएका छन् । A box contains 6 white and 10 black balls of same shape and size. Two balls are drawn at random one after another with replacement. (a) यदि A र B दुईओटा अनाश्रित घट्नाहरू भए सम्भाव्यताको गुणन सिद्धान्त लेर्नुहोस् । If A and B are two independent events, write the multiplication law of probability. (b) सबै सम्भावित परिणामहरूको सम्भाव्यतालाई वृक्षचित्रमा देखाउनुहोस् । Show the probability of all the possible outcomes in a tree diagram. (2) (c) दुवै बल उही रङ्का पर्ने सम्भाव्यता पत्ता लगाउनुहोस् । Find the probability of getting both balls of same color. (d) दुवै बल फरक रडका पर्ने सम्भाव्यता, दुवै बल सेता रडका पर्ने सम्भाव्यता भन्दा कति कम वा बढी हुन्छ ? पत्ता लगाउनुहोस् । By how much the the probability of getting both balls of different color is less or more than probability of getting both balls of white color? Find it. (1)

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### Given:
- Total white balls = 6
- Total black balls = 10
- Total balls = 6 + 10 = 16

### (a) Multiplication Law of Probability
If $$ A $$ and $$ B $$ are two independent events, the multiplication law of probability states that:
$$
P(A \cap B) = P(A) \times P(B)
$$
This means the probability of both events $$ A $$ and $$ B $$ occurring is the product of their individual probabilities.

### (b) Tree Diagram for All Possible Outcomes
To represent the outcomes in a tree diagram:

1. **First Draw:**
   - White (W)
   - Black (B)

2. **Second Draw (after replacement):**
   - If the first draw was White (W):
     - White (W)
     - Black (B)
   - If the first draw was Black (B):
     - White (W)
     - Black (B)

The tree diagram can be represented as follows:

```
          First Draw
          /      \
         W        B
        / \      / \
       W   B    W   B
```

### Probabilities of Outcomes:
- Probability of drawing a white ball $$ P(W) = \frac{6}{16} = \frac{3}{8} $$
- Probability of drawing a black ball $$ P(B) = \frac{10}{16} = \frac{5}{8} $$

The probabilities for each branch of the tree are:
- $$ P(W, W) = P(W) \times P(W) = \frac{3}{8} \times \frac{3}{8} = \frac{9}{64} $$
- $$ P(W, B) = P(W) \times P(B) = \frac{3}{8} \times \frac{5}{8} = \frac{15}{64} $$
- $$ P(B, W) = P(B) \times P(W) = \frac{5}{8} \times \frac{3}{8} = \frac{15}{64} $$
- $$ P(B, B) = P(B) \times P(B) = \frac{5}{8} \times \frac{5}{8} = \frac{25}{64} $$

### (c) Probability of Getting Both Balls of Same Color
The events of getting both balls of the same color are:
1. Both White: $$ P(W, W) = \frac{9}{64} $$
2. Both Black: $$ P(B, B) = \frac{25}{64} $$

Thus, the total probability of getting both balls of the same color is:
$$
P(\text{Same Color}) = P(W, W) + P(B, B) = \frac{9}{64} + \frac{25}{64} = \frac{34}{64} = \frac{17}{32}
$$

### (d) Probability of Getting Both Balls of Different Colors
The events of getting both balls of different colors are:
1. First White, Second Black: $$ P(W, B) = \frac{15}{64} $$
2. First Black, Second White: $$ P(B, W) = \frac{15}{64} $$

Thus, the total probability of getting both balls of different colors is:
$$
P(\text{Different Color}) = P(W, B) + P(B, W) = \frac{15}{64} + \frac{15}{64} = \frac{30}{64} = \frac{15}{32}
$$

### Comparison
Now, we compare the probabilities:
- Probability of both balls of white color: $$ P(W, W) = \frac{9}{64} $$
- Probability of both balls of different colors: $$ P(\text{Different Color}) = \frac{15}{32} = \frac{30}{64} $$

To find the difference:
$$
P(\text{Different Color}) - P(W, W) = \frac{30}{64} - \frac{9}{64} = \frac{21}{64}
$$

### Summary of Results:
- (a) Multiplication Law: $$ P(A \cap B) = P(A) \times P(B) $$
- (b) Tree Diagram: Represented above with probabilities.
- (c) Probability of same color: $$ \frac{17}{32} $$
- (d) Difference in probabilities: $$ \frac{21}{64} $$ (Probability of different colors is more than that of both white balls).
(a) The probability of both events A and B occurring is the product of their individual probabilities: $$ P(A \cap B) = P(A) \times P(B) $$. (b) The tree diagram shows all possible outcomes with their probabilities: - First draw: White (W) or Black (B) - Second draw: White (W) or Black (B) - Probabilities: $$ \frac{9}{64} $$ for both white, $$ \frac{15}{64} $$ for white then black, $$ \frac{15}{64} $$ for black then white, and $$ \frac{25}{64} $$ for both black. (c) The probability of getting both balls of the same color is $$ \frac{17}{32} $$. (d) The probability of getting both balls of different colors is $$ \frac{15}{32} $$, which is $$ \frac{21}{64} $$ more than the probability of getting both white balls.

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Solución

Given:

(a) Multiplication Law of Probability

(b) Tree Diagram for All Possible Outcomes

Probabilities of Outcomes:

© Probability of Getting Both Balls of Same Color

(d) Probability of Getting Both Balls of Different Colors

Comparison

Summary of Results:

The Deep Dive

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