If three coins are tossed simultaneously, find the probability of getting:
(i) two heads
(ii) one head and two tails
(iii) at least one head

Explanation

To solve the problem of finding the probability of different outcomes when three coins are tossed simultaneously, we can follow these steps:

### Step 1: Identify the Sample Space
When three coins are tossed, 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 sample space \( S \) consists of 8 outcomes:
\[ S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \} \]

### Step 2: Calculate the Probability of Getting Two Heads
Let \( E \) be the event of getting two heads. The favorable outcomes for this event are:
- HHT
- HTH
- THH

So, there are 3 favorable outcomes for event \( E \).

The probability \( P(E) \) is given by:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8} \]

### Step 3: Calculate the Probability of Getting One Head and Two Tails
Let \( F \) be the event of getting one head and two tails. The favorable outcomes for this event are:
- HTT
- THT
- TTH

So, there are 3 favorable outcomes for event \( F \).

The probability \( P(F) \) is given by:
\[ P(F) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8} \]

### Step 4: Calculate the Probability of Getting At Least One Head
Let \( G \) be the event of getting at least one head. The favorable outcomes for this event are:
- HHH
- HHT
- HTH
- THH
- HTT
- THT
- TTH

The only outcome that does not have at least one head is TTT. Therefore, there are 7 favorable outcomes for event \( G \).

The probability \( P(G) \) is given by:
\[ P(G) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{7}{8} \]

### Final Answers
- (i) Probability of getting two heads: \( \frac{3}{8} \)
- (ii) Probability of getting one head and two tails: \( \frac{3}{8} \)
- (iii) Probability of getting at least one head: \( \frac{7}{8} \)

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