We are tossing an unbiased coin three times. For each toss, the probability of getting a head (H) or a tail (T) is $\frac{1}{2}$. The total number of possible outcomes for three tosses is $2^3 = 8$. These outcomes are:

$\{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \}$

(i) Probability of getting one head:

The outcomes with exactly one head are $HTT, THT, TTH$. There are 3 such outcomes.

The probability of getting exactly one head is:

$P(\text{one head}) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}} = \frac{3}{8}$

(ii) Probability of getting the first head:

For this, we need the first coin to show a head. The possible outcomes are those that start with $H$, which are $HHH, HHT, HTH, HTT$. There are 4 such outcomes.

Thus, the probability of getting the first head is:

$P(\text{first head}) = \frac{4}{8} = \frac{1}{2}$

(iii) Probability of getting at least one head:

To find the probability of getting at least one head, we can first find the probability of getting no heads (i.e., all tails). The outcome where no heads appear is $TTT$.

So, the probability of getting no heads is:

$P(\text{no heads}) = \frac{1}{8}$

Thus, the probability of getting at least one head is:

$P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - \frac{1}{8} = \frac{7}{8}$

(iv) Probability of heads and tails alternating:

The outcomes where heads and tails alternate are $HTH$ and $THT$. There are 2 such outcomes.

Thus, the probability of getting heads and tails in an alternating order is:

$P(\text{alternating heads and tails}) = \frac{2}{8} = \frac{1}{4}$

Summary:

1. One head: $\frac{3}{8}$
2. First head: $\frac{1}{2}$
3. At least one head: $\frac{7}{8}$
4. Heads and tails alternate: $\frac{1}{4}$

Would you like more details or have any questions?

5 Related Questions:

1. What is the probability of getting two heads in three tosses?
2. What is the probability of getting no tails at all in three tosses?
3. How does the probability change for a biased coin where the probability of heads is $\frac{3}{4}$?
4. What is the expected number of heads in three coin tosses?
5. What is the probability of getting exactly one head and two tails?

Tip:

Remember that in probability, it is often easier to calculate the complement (the opposite event) if that simplifies the calculation.