wo unbiased coins are tossed simultaneously . Find the probability of getting(I)no heads, (ii)at most one tail(Iii)One tail (iv) one head and one tail

1. Break Down the Problem

We need to find the probabilities for four different scenarios when two unbiased coins are tossed:

1. No heads
2. At most one tail
3. One tail
4. One head and one tail

2. Relevant Concepts

When two unbiased coins are tossed, the sample space $S$ consists of all the possible outcomes. The possible outcomes when tossing two coins are:

• HH (both heads)
• HT (one head, one tail)
• TH (one head, one tail)
• TT (both tails)

Thus, the sample space can be expressed as: $S = \{ HH, HT, TH, TT \}$

The total number of outcomes $n(S)$ is 4.

3. Analysis and Detail

We will analyze each of the four scenarios mentioned:

1. No Heads (TT)

   • Outcomes: TT
   • Number of favorable outcomes: 1
   • Probability: $P(\text{No Heads}) = \frac{1}{4}$

2. At Most One Tail

   • Outcomes: HT, TH, TT
   • Number of favorable outcomes: 3
   • Probability: $P(\text{At Most One Tail}) = \frac{3}{4}$

3. One Tail (HT, TH)

   • Outcomes: HT, TH
   • Number of favorable outcomes: 2
   • Probability: $P(\text{One Tail}) = \frac{2}{4} = \frac{1}{2}$

4. One Head and One Tail (HT, TH)

   • Outcomes: HT, TH
   • Number of favorable outcomes: 2
   • Probability: $P(\text{One Head and One Tail}) = \frac{2}{4} = \frac{1}{2}$

4. Verify and Summarize

Each calculation has been verified based on the total possible outcomes. The probabilities calculated are consistent with the favorable outcomes for each scenario.

Final Answer

1. No heads: $P = \frac{1}{4}$
2. At most one tail: $P = \frac{3}{4}$
3. One tail: $P = \frac{1}{2}$
4. One head and one tail: $P = \frac{1}{2}$