In the image you provided, there is a probability table that relates to a scenario where a coin is tossed three times, and different outcomes (HHH, TTT, etc.) are analyzed based on three specific events (A, B, and C). The task is to check which outcomes correspond to each event and calculate the probability of the event occurring.

Let’s break down each event:

1. Event A: A tail on both the first and the last tosses.

   • You are looking for outcomes where the first and the third tosses are both tails.
   • Outcomes that fit: TTT, THT.

2. Event B: Two or more heads.

   • You are looking for outcomes where there are at least two heads (H).
   • Outcomes that fit: HHH, HHT, THH.

3. Event C: A tail on the first toss or the third toss (or both).

   • You are looking for outcomes where the first or the third toss (or both) are tails.
   • Outcomes that fit: TTT, TTH, THT, HTT.

Probability Calculation:

• Since there are 8 possible outcomes (HHH, TTT, THT, THH, HTH, HHT, HTT, TTH), the probability of each event is calculated by dividing the number of outcomes satisfying the event by 8.

1. Event A:

   • Outcomes: TTT, THT (2 outcomes)
   • Probability: $\frac{2}{8} = 0.25$

2. Event B:

   • Outcomes: HHH, HHT, THH (3 outcomes)
   • Probability: $\frac{3}{8} = 0.375$

3. Event C:

   • Outcomes: TTT, TTH, THT, HTT (4 outcomes)
   • Probability: $\frac{4}{8} = 0.5$

Let me know if you want further clarification or additional details on this! Here's a set of related questions to expand on the concept:

1. How would the probabilities change if the coin was biased towards heads or tails?
2. What is the probability of getting exactly one head in three tosses?
3. Can you compute the probability of getting no heads in a series of four tosses?
4. How would the probabilities differ if you tossed the coin four times instead of three?
5. What is the complement of Event B (i.e., having less than two heads)?

Tip: When calculating probabilities for independent events like coin tosses, the total number of outcomes is determined by multiplying the possible outcomes for each event (in this case, $2^n$ for n tosses).