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 will follow these steps:

### Step 1: Determine the Sample Space
When three coins are tossed, each coin can either land as Heads (H) or Tails (T). The total number of outcomes can be calculated as follows:

- Each coin has 2 possible outcomes (H or T).
- Therefore, for 3 coins, the total number of outcomes is \(2^3 = 8\).

The sample space (S) of all possible outcomes is:
\[ S = \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \]

### Step 2: Calculate the Probability of Getting Two Heads
Next, we need to find the number of favorable outcomes for getting exactly two heads.

The favorable outcomes for getting exactly two heads are:
- HHT
- HTH
- THH

Thus, there are 3 favorable outcomes.

Now, we can calculate the probability:
\[ P(\text{Two Heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} \]

### Step 3: Calculate the Probability of Getting One Head and Two Tails
Now, we find the number of favorable outcomes for getting exactly one head and two tails.

The favorable outcomes for getting exactly one head are:
- HTT
- THT
- TTH

Thus, there are also 3 favorable outcomes.

Now, we can calculate the probability:
\[ P(\text{One Head and Two Tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} \]

### Step 4: Calculate the Probability of Getting At Least One Head
To find the probability of getting at least one head, we can use the complement rule. First, we find the probability of getting no heads (which means getting all tails).

The only outcome with no heads is:
- TTT

Thus, there is 1 favorable outcome for getting no heads.

Now, we can calculate the probability of getting no heads:
\[ P(\text{No Heads}) = \frac{1}{8} \]

Therefore, the probability of getting at least one head is:
\[ P(\text{At Least One Head}) = 1 - P(\text{No Heads}) = 1 - \frac{1}{8} = \frac{7}{8} \]

### 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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