If two coins are tossed once , what is the probability of getting (i) both heads , (ii) both heads or both talls (iii) at least one head ?

Answer

To solve the problem of finding the probability of certain outcomes when two coins are tossed, we will follow these steps:

### Step 1: Identify the Sample Space
When two coins are tossed, the possible outcomes are:
1. HH (both heads)
2. HT (head on the first coin, tail on the second)
3. TH (tail on the first coin, head on the second)
4. TT (both tails)

Thus, the sample space \( S \) is:
\[ S = \{ HH, HT, TH, TT \} \]

### Step 2: Calculate Total Outcomes
The total number of outcomes when two coins are tossed is:
\[ \text{Total Outcomes} = 4 \]

### Step 3: Calculate Probability of Getting Both Heads
(i) We need to find the probability of getting both heads (HH).
- **Favorable Outcomes**: There is 1 favorable outcome (HH).
- **Total Outcomes**: There are 4 total outcomes.

Using the probability formula:
\[ P(\text{Both Heads}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{1}{4} \]

### Step 4: Calculate Probability of Getting Both Heads or Both Tails
(ii) Now, we need to find the probability of getting both heads (HH) or both tails (TT).
- **Favorable Outcomes**: There are 2 favorable outcomes (HH and TT).
- **Total Outcomes**: There are 4 total outcomes.

Using the probability formula:
\[ P(\text{Both Heads or Both Tails}) = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Calculate Probability of Getting At Least One Head
(iii) Finally, we need to find the probability of getting at least one head.
- **Favorable Outcomes**: The favorable outcomes for at least one head are (HH, HT, TH). There are 3 favorable outcomes.
- **Total Outcomes**: There are 4 total outcomes.

Using the probability formula:
\[ P(\text{At Least One Head}) = \frac{3}{4} \]

### Summary of Results
- (i) Probability of both heads: \( \frac{1}{4} \)
- (ii) Probability of both heads or both tails: \( \frac{1}{2} \)
- (iii) Probability of at least one head: \( \frac{3}{4} \)

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