A coin is tossed once. Then ifit turns up a head a die is thrown once, if it turns up a tail it is tossed twice more. Describe:
(i) the sample space S of the experiment
(ii) the event A that exactly one head occurs.
(iii) the event B that atleast two tails occur or a number greater than 4 occurs.

Answer

To solve the problem step by step, let's break it down into parts as requested:

### (i) Sample Space S of the Experiment

1. **Toss the Coin**: The first step is to toss a coin. The outcomes can be either Head (H) or Tail (T).
2. **If Head (H)**: If the outcome is Head, we then roll a die. The possible outcomes from rolling a die are 1, 2, 3, 4, 5, or 6. Therefore, the outcomes when the coin shows Head are:
- H1 (Head and die shows 1)
- H2 (Head and die shows 2)
- H3 (Head and die shows 3)
- H4 (Head and die shows 4)
- H5 (Head and die shows 5)
- H6 (Head and die shows 6)

3. **If Tail (T)**: If the outcome is Tail, we toss the coin two more times. The possible outcomes when tossing the coin twice are:
- HH (Head, Head)
- HT (Head, Tail)
- TH (Tail, Head)
- TT (Tail, Tail)

4. **Combine Outcomes**: Now, we combine the outcomes from both scenarios:
- From Head: {H1, H2, H3, H4, H5, H6}
- From Tail: {T, HH, HT, TH, TT}

Thus, the complete sample space S is:
\[ S = \{ H1, H2, H3, H4, H5, H6, T, HH, HT, TH, TT \} \]

### (ii) Event A: Exactly One Head Occurs

1. **Identify Outcomes with Exactly One Head**: We need to find the outcomes in the sample space where exactly one Head occurs.
2. **From the outcomes of H**: All outcomes from rolling the die (H1, H2, H3, H4, H5, H6) contain exactly one Head.
3. **From the outcomes of T**: The outcomes that contain exactly one Head when tossing the coin twice are:
- HT (Head, Tail)
- TH (Tail, Head)

4. **Combine Outcomes**: Therefore, the event A can be represented as:
\[ A = \{ H1, H2, H3, H4, H5, H6, HT, TH \} \]

### (iii) Event B: At Least Two Tails Occur or a Number Greater than 4 Occurs

1. **Identify Outcomes with At Least Two Tails**: We need to find outcomes in the sample space where there are at least two Tails.
- From the outcomes of T:
- TT (Tail, Tail)
- HT (Head, Tail) does not count as it has only one Tail.
- TH (Tail, Head) does not count as it has only one Tail.
- HH (Head, Head) does not count as it has no Tails.
- Therefore, the only outcome with at least two Tails is TT.

2. **Identify Outcomes with a Number Greater than 4**: The outcomes from rolling the die that are greater than 4 are:
- H5 (Head and die shows 5)
- H6 (Head and die shows 6)

3. **Combine Outcomes**: Therefore, the event B can be represented as:
\[ B = \{ TT, H5, H6 \} \]

### Final Answers
- Sample Space \( S = \{ H1, H2, H3, H4, H5, H6, T, HH, HT, TH, TT \} \)
- Event \( A = \{ H1, H2, H3, H4, H5, H6, HT, TH \} \)
- Event \( B = \{ TT, H5, H6 \} \)