Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes.
`|{:("Outcome","3 heads","2 heads","1 head","No head"),("Frequency"," 23"," 72"," 77"," 28"):}|`
Find the probabiltiy of (i) getting at most two heads (ii) getting at least two heads.

Answer

To solve the problem step by step, we will calculate the probabilities for both parts of the question.

### Given Data:
- Total tosses (N) = 200
- Frequencies of outcomes:
- 3 heads: 23
- 2 heads: 72
- 1 head: 77
- No head: 28

### Part (i): Probability of getting at most two heads

1. **Identify the outcomes for at most two heads**:
- At most two heads means we can have either 0 heads, 1 head, or 2 heads.
- The frequencies for these outcomes are:
- No head (0 heads): 28
- 1 head: 77
- 2 heads: 72

2. **Calculate the total favorable outcomes for at most two heads**:
\[ \text{Favorable outcomes} = \text{Frequency of 0 heads} + \text{Frequency of 1 head} + \text{Frequency of 2 heads} \]
\[ = 28 + 77 + 72 = 177 \]

3. **Calculate the probability**:
\[ P(\text{at most 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{177}{200} \]

### Part (ii): Probability of getting at least two heads

1. **Identify the outcomes for at least two heads**:
- At least two heads means we can have either 2 heads or 3 heads.
- The frequencies for these outcomes are:
- 2 heads: 72
- 3 heads: 23

2. **Calculate the total favorable outcomes for at least two heads**:
\[ \text{Favorable outcomes} = \text{Frequency of 2 heads} + \text{Frequency of 3 heads} \]
\[ = 72 + 23 = 95 \]

3. **Calculate the probability**:
\[ P(\text{at least 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{95}{200} \]
\[ = \frac{19}{40} \quad \text{(after simplifying)} \]

### Final Answers:
- (i) The probability of getting at most two heads is \(\frac{177}{200}\).
- (ii) The probability of getting at least two heads is \(\frac{19}{40}\).