Three unbiased coins are tossed once. Find the probability of getting:
(i) two heads (ii) one head or two heads.
(iii) at least 2 heads
(iv) at most 2 heads
v. at most one head.

Answer

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

### Step 1: Determine the Sample Space
When tossing three unbiased coins, each coin can either be Heads (H) or Tails (T). The total number of outcomes when tossing three coins is calculated as:
\[ 2^3 = 8 \]
The sample space (S) consists of the following outcomes:
- HHH
- HHT
- HTH
- THH
- HTT
- THT
- TTH
- TTT

### Step 2: Calculate the Probability for Each Part

#### (i) Probability of getting two heads
To find the probability of getting exactly two heads, we need to count the outcomes that have exactly two heads:
- HHT
- HTH
- THH

There are 3 outcomes with exactly two heads. Therefore, the probability \( P(2 \text{ heads}) \) is:
\[ P(2 \text{ heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} \]

#### (ii) Probability of getting one head or two heads
Next, we need to find the outcomes with either one head or two heads:
- Outcomes with one head: HTT, THT, TTH (3 outcomes)
- Outcomes with two heads: HHT, HTH, THH (3 outcomes)

Total outcomes with one or two heads = 3 + 3 = 6. Therefore, the probability \( P(1 \text{ head or } 2 \text{ heads}) \) is:
\[ P(1 \text{ head or } 2 \text{ heads}) = \frac{6}{8} = \frac{3}{4} \]

#### (iii) Probability of getting at least 2 heads
At least 2 heads means we can have either 2 heads or 3 heads. The outcomes are:
- Outcomes with two heads: HHT, HTH, THH (3 outcomes)
- Outcomes with three heads: HHH (1 outcome)

Total outcomes with at least 2 heads = 3 + 1 = 4. Therefore, the probability \( P(\text{at least } 2 \text{ heads}) \) is:
\[ P(\text{at least } 2 \text{ heads}) = \frac{4}{8} = \frac{1}{2} \]

#### (iv) Probability of getting at most 2 heads
At most 2 heads means we can have 0, 1, or 2 heads. The outcomes are:
- Outcomes with zero heads: TTT (1 outcome)
- Outcomes with one head: HTT, THT, TTH (3 outcomes)
- Outcomes with two heads: HHT, HTH, THH (3 outcomes)

Total outcomes with at most 2 heads = 1 + 3 + 3 = 7. Therefore, the probability \( P(\text{at most } 2 \text{ heads}) \) is:
\[ P(\text{at most } 2 \text{ heads}) = \frac{7}{8} \]

#### (v) Probability of getting at most one head
At most one head means we can have 0 or 1 head. The outcomes are:
- Outcomes with zero heads: TTT (1 outcome)
- Outcomes with one head: HTT, THT, TTH (3 outcomes)

Total outcomes with at most one head = 1 + 3 = 4. Therefore, the probability \( P(\text{at most } 1 \text{ head}) \) is:
\[ P(\text{at most } 1 \text{ head}) = \frac{4}{8} = \frac{1}{2} \]

### Summary of Probabilities
- (i) Probability of two heads: \( \frac{3}{8} \)
- (ii) Probability of one head or two heads: \( \frac{3}{4} \)
- (iii) Probability of at least 2 heads: \( \frac{1}{2} \)
- (iv) Probability of at most 2 heads: \( \frac{7}{8} \)
- (v) Probability of at most one head: \( \frac{1}{2} \)