In a simultaneously toss of two coins, find the probability of:
(i) exactl 2 tails (ii) exactly 1 tail (iii) no tails.

Answer

To solve the problem of finding the probability of getting exactly 2 tails, exactly 1 tail, and no tails when tossing two coins, we can follow these steps:

### Step 1: Determine the Sample Space
When tossing two coins, each coin has two possible outcomes: Heads (H) or Tails (T). Therefore, the total number of outcomes when tossing two coins can be calculated as:

\[ \text{Total Outcomes} = 2^n = 2^2 = 4 \]

The sample space (S) of the outcomes is:
- HH (both heads)
- HT (one head, one tail)
- TH (one tail, one head)
- TT (both tails)

So, the sample space is: S = {HH, HT, TH, TT}

### Step 2: Calculate the Probability of Exactly 2 Tails
To find the probability of getting exactly 2 tails (TT), we count the number of favorable outcomes for this event.

- Favorable outcomes for exactly 2 tails: 1 (TT)

Thus, the probability (P) of getting exactly 2 tails is:

\[ P(\text{exactly 2 tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{4} \]

### Step 3: Calculate the Probability of Exactly 1 Tail
Next, we find the probability of getting exactly 1 tail. The favorable outcomes for this event are:

- HT (one head, one tail)
- TH (one tail, one head)

Thus, the number of favorable outcomes for exactly 1 tail is 2.

So, the probability of getting exactly 1 tail is:

\[ P(\text{exactly 1 tail}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{4} = \frac{1}{2} \]

### Step 4: Calculate the Probability of No Tails
Finally, we find the probability of getting no tails. The only outcome that has no tails is:

- HH (both heads)

Thus, the number of favorable outcomes for no tails is 1.

So, the probability of getting no tails is:

\[ P(\text{no tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{4} \]

### Summary of Probabilities
- Probability of exactly 2 tails: \( \frac{1}{4} \)
- Probability of exactly 1 tail: \( \frac{1}{2} \)
- Probability of no tails: \( \frac{1}{4} \)