Three coins are tossed simultaneously. Consider the event E 'three heads or three tails' F 'at least two heads' and G 'at the most two heads'. Of the pairs (E, F), (E, G) and (F, G)
(i) Which are independent?, `" "` (ii) Which are dependent?

Answer

To solve the problem, we need to analyze the events E, F, and G based on the outcomes of tossing three coins. Let's break down the solution step by step.

### Step 1: Determine the Sample Space
When three coins are tossed, the total number of outcomes is given by \(2^3 = 8\). The sample space \(S\) is:
\[ S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \} \]
where H represents heads and T represents tails.

### Step 2: Define the Events
- **Event E**: "Three heads or three tails"
- Outcomes: \( E = \{ HHH, TTT \} \)

- **Event F**: "At least two heads"
- Outcomes: \( F = \{ HHH, HHT, HTH, THH \} \)

- **Event G**: "At most two heads"
- Outcomes: \( G = \{ HHT, HTH, THH, HTT, THT, TTH, TTT \} \)

### Step 3: Calculate the Probabilities of Each Event
- Total outcomes = 8

- Probability of E:
\[ P(E) = \frac{|E|}{|S|} = \frac{2}{8} = \frac{1}{4} \]

- Probability of F:
\[ P(F) = \frac{|F|}{|S|} = \frac{4}{8} = \frac{1}{2} \]

- Probability of G:
\[ P(G) = \frac{|G|}{|S|} = \frac{7}{8} \]

### Step 4: Find Intersections of Events
- **Intersection of E and F**:
\[ E \cap F = \{ HHH \} \quad \Rightarrow \quad P(E \cap F) = \frac{1}{8} \]

- **Intersection of E and G**:
\[ E \cap G = \{ TTT \} \quad \Rightarrow \quad P(E \cap G) = \frac{1}{8} \]

- **Intersection of F and G**:
\[ F \cap G = \{ HHT, HTH, THH \} \quad \Rightarrow \quad P(F \cap G) = \frac{3}{8} \]

### Step 5: Check for Independence
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]

- **Check Independence of E and F**:
\[ P(E) \cdot P(F) = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]
Since \(P(E \cap F) = \frac{1}{8}\), E and F are independent.

- **Check Independence of E and G**:
\[ P(E) \cdot P(G) = \frac{1}{4} \cdot \frac{7}{8} = \frac{7}{32} \]
Since \(P(E \cap G) = \frac{1}{8} = \frac{4}{32}\), E and G are dependent.

- **Check Independence of F and G**:
\[ P(F) \cdot P(G) = \frac{1}{2} \cdot \frac{7}{8} = \frac{7}{16} \]
Since \(P(F \cap G) = \frac{3}{8} = \frac{6}{16}\), F and G are dependent.

### Final Results
(i) **Independent Events**: (E, F)
(ii) **Dependent Events**: (E, G) and (F, G)