Three unbiased coins are tossed together. Find the probability of getting:
(i) all heads.
(ii) exactly two heads.
(iii) exactly one head.
(iv) at least two heads.
(v) at least two tails

Answer

To solve the problem of finding the probabilities of different outcomes when three unbiased coins are tossed, we will follow these steps:

### Step 1: Determine the Sample Space
When three unbiased coins are tossed, the possible outcomes (sample space) can be represented as:
- 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 number of outcomes in the sample space is 8.

### Step 2: Calculate Probability of All Heads
The event of getting all heads (HHH) has only 1 favorable outcome.
- Probability of all heads = Number of favorable outcomes / Total outcomes = 1 / 8.

### Step 3: Calculate Probability of Exactly Two Heads
The outcomes with exactly two heads are HHT, HTH, and THH. There are 3 favorable outcomes.
- Probability of exactly two heads = Number of favorable outcomes / Total outcomes = 3 / 8.

### Step 4: Calculate Probability of Exactly One Head
The outcomes with exactly one head are HTT, THT, and TTH. There are also 3 favorable outcomes.
- Probability of exactly one head = Number of favorable outcomes / Total outcomes = 3 / 8.

### Step 5: Calculate Probability of At Least Two Heads
The outcomes with at least two heads include HHH, HHT, HTH, and THH. There are 4 favorable outcomes.
- Probability of at least two heads = Number of favorable outcomes / Total outcomes = 4 / 8 = 1 / 2.

### Step 6: Calculate Probability of At Least Two Tails
The outcomes with at least two tails include TTT, TTH, THT, and HTT. There are also 4 favorable outcomes.
- Probability of at least two tails = Number of favorable outcomes / Total outcomes = 4 / 8 = 1 / 2.

### Summary of Probabilities
1. Probability of all heads = 1/8
2. Probability of exactly two heads = 3/8
3. Probability of exactly one head = 3/8
4. Probability of at least two heads = 1/2
5. Probability of at least two tails = 1/2