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)

Question

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### 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.

Upstudy AI Solution

Answer

Solution

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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