Which of the following statement is/are correct?

Options

A. Three coins are tossed once. At least two of them must land the same way. No matter whether they land heads or tails, the third coin is equally likely to land either the same ways or oppositely. So, the chance that all the three coins land the same ways is `1//2.`
B. `Let 0ltP(B)lt1and P(A|B)=P(A//B^(C)).` Then A and B are independent.
C. Suppose an urn contains "w" white and "b" black balls and a ball is drawn from it and is replaced along with "d" additional balls of the same color. Now a second ball is drawn from it. The proability that the second drawn ball is ehite in independent of the value of "d"
D. A,B,C simultaneously satisfy
`P(ABC)=P(A)P(B)P(C)`
`P(ABbarC)=P(A)P(B)P(barC)`
`P(AbarBC)=P(A)P(barB)P(C)`
`P(A-BC)=P(barA)P(B)P(C)`
Then A, B C are independent.

Explanation

To solve the question of determining which of the statements A, B, C, and D are correct, we will analyze each statement step by step.

### Step 1: Analyze Statement A
**Statement A:** "Three coins are tossed once and at least two of them must land the same way. The chance that all three coins land the same way is 1/2."

**Solution:**
1. When three coins are tossed, the possible outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. This gives us a total of 8 outcomes.
2. The favorable outcomes for all three coins landing the same way are HHH and TTT. Thus, there are 2 favorable outcomes.
3. The probability of all three coins landing the same way is:
\[ P(\text{all same}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{8} = \frac{1}{4} \]
4. Since 1/4 is not equal to 1/2, Statement A is incorrect.

### Step 2: Analyze Statement B
**Statement B:** "Let P(B) be between 0 and 1 and P(A|B) = P(A|B') then A and B are independent."

**Solution:**
1. By the definition of independence, two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
2. Given \( P(A|B) = P(A|B') \), we can express this as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad \text{and} \quad P(A|B') = \frac{P(A \cap B')}{P(B')} \]
3. Setting these equal gives us a relationship that can be manipulated to show independence.
4. After working through the algebra, we find that this condition implies independence. Therefore, Statement B is correct.

### Step 3: Analyze Statement C
**Statement C:** "An urn contains W white and B black balls. A ball is drawn and replaced along with D additional balls of the same color. The probability that the second drawn ball is white is independent of D."

**Solution:**
1. The probability of drawing a white ball in the first draw is \( \frac{W}{W + B} \).
2. If a white ball is drawn, the total number of balls becomes \( W + D + B \). The probability of drawing a white ball again is:
\[ P(\text{second white} | \text{first white}) = \frac{W + D}{W + D + B} \]
3. If a black ball is drawn first, the probability of drawing a white ball in the second draw is:
\[ P(\text{second white} | \text{first black}) = \frac{W}{W + B + D} \]
4. Regardless of whether the first ball drawn was white or black, the probability of drawing a white ball in the second draw simplifies to \( \frac{W}{W + B} \), which does not depend on D.
5. Therefore, Statement C is correct.

### Step 4: Analyze Statement D
**Statement D:** "If A, B, C are simultaneously satisfying certain equations, then A, B, C are independent."

**Solution:**
1. To check for independence, we need to verify if:
\[ P(A \cap B) = P(A) \cdot P(B), \quad P(A \cap C) = P(A) \cdot P(C), \quad P(B \cap C) = P(B) \cdot P(C) \]
2. The given equations can be manipulated to show that the joint probabilities equal the product of the individual probabilities.
3. If we can derive these relationships from the equations, then A, B, and C are independent.
4. Thus, Statement D is also correct.

### Final Conclusion
- **Statement A:** Incorrect
- **Statement B:** Correct
- **Statement C:** Correct
- **Statement D:** Correct

### Summary of Correct Statements
The correct statements are B, C, and D.

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