A coin is tossed three times. Find the probability of getting:
(i) all tails. (ii) at least two tails. (iii) atmost two tails.

Answer

To solve the problem of finding the probabilities when a coin is tossed three times, we will follow these steps:

### Step 1: Determine the Sample Space
When a coin is tossed three times, the possible outcomes can be represented as follows:
- HHH (3 Heads)
- HHT (2 Heads, 1 Tail)
- HTH (2 Heads, 1 Tail)
- THH (2 Heads, 1 Tail)
- HTT (1 Head, 2 Tails)
- THT (1 Head, 2 Tails)
- TTH (1 Head, 2 Tails)
- TTT (3 Tails)

Thus, the total sample space (S) consists of 8 outcomes:
\[ S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \} \]

### Step 2: Calculate the Probability of Getting All Tails
To find the probability of getting all tails (TTT):
- Favorable outcomes = 1 (only TTT)
- Total outcomes = 8

The probability (P) is given by:
\[ P(\text{All Tails}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{1}{8} \]

### Step 3: Calculate the Probability of Getting At Least Two Tails
To find the probability of getting at least two tails, we consider the outcomes that have 2 or more tails:
- Favorable outcomes = HTT, THT, TTH, TTT (4 outcomes)

Thus:
\[ P(\text{At Least 2 Tails}) = \frac{4}{8} = \frac{1}{2} \]

### Step 4: Calculate the Probability of Getting At Most Two Tails
To find the probability of getting at most two tails, we consider the outcomes that have 0, 1, or 2 tails:
- Favorable outcomes = HHH, HHT, HTH, THH, HTT, THT, TTH (7 outcomes)

Thus:
\[ P(\text{At Most 2 Tails}) = \frac{7}{8} \]

### Summary of Results
1. Probability of getting all tails: \( \frac{1}{8} \)
2. Probability of getting at least two tails: \( \frac{1}{2} \)
3. Probability of getting at most two tails: \( \frac{7}{8} \)

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