(a) A coin is tossed three times.
(i) E : heads on third toss , F : heads on first two tosses
(ii) E : at least two heads,
F : at most two heads
(iii) E : at most two tails , F : at least one tail.
Find the probability in all cases.

Answer

To solve the problem, we will analyze each part step by step, calculating the probabilities for events E and F as described in the question.

### Total Outcomes
When a coin is tossed three times, the total number of outcomes can be calculated as follows:

- Each toss has 2 possible outcomes: Heads (H) or Tails (T).
- Therefore, for 3 tosses, the total number of outcomes is:
\[ 2^3 = 8 \]

The possible outcomes are:
1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT

### Part (i)
**E:** Heads on the third toss
**F:** Heads on the first two tosses

1. **Calculate P(E):**
The outcomes where the third toss is a head are:
- HHH
- HHT
- HTH
- HTT

Thus, there are 4 favorable outcomes for E.
\[ P(E) = \frac{\text{Number of favorable outcomes for E}}{\text{Total outcomes}} = \frac{4}{8} = \frac{1}{2} \]

2. **Calculate P(F):**
The outcomes where the first two tosses are heads are:
- HHH
- HHT

Thus, there are 2 favorable outcomes for F.
\[ P(F) = \frac{\text{Number of favorable outcomes for F}}{\text{Total outcomes}} = \frac{2}{8} = \frac{1}{4} \]

### Part (ii)
**E:** At least two heads
**F:** At most two heads

1. **Calculate P(E):**
The outcomes with at least two heads are:
- HHH
- HHT
- HTH
- THH

Thus, there are 4 favorable outcomes for E.
\[ P(E) = \frac{4}{8} = \frac{1}{2} \]

2. **Calculate P(F):**
The outcomes with at most two heads are:
- HHT
- HTH
- THH
- HTT
- THT
- TTH
- TTT

Thus, there are 7 favorable outcomes for F.
\[ P(F) = \frac{7}{8} \]

### Part (iii)
**E:** At most two tails
**F:** At least one tail

1. **Calculate P(E):**
The outcomes with at most two tails are:
- HHH
- HHT
- HTH
- THH
- HTT
- THT
- TTH

Thus, there are 7 favorable outcomes for E.
\[ P(E) = \frac{7}{8} \]

2. **Calculate P(F):**
The outcomes with at least one tail are:
- HHT
- HTH
- HTT
- THH
- THT
- TTH
- TTT

Thus, there are 7 favorable outcomes for F.
\[ P(F) = \frac{7}{8} \]

### Summary of Probabilities
- **Part (i):**
- \( P(E) = \frac{1}{2} \)
- \( P(F) = \frac{1}{4} \)

- **Part (ii):**
- \( P(E) = \frac{1}{2} \)
- \( P(F) = \frac{7}{8} \)

- **Part (iii):**
- \( P(E) = \frac{7}{8} \)
- \( P(F) = \frac{7}{8} \)