An unbiased coin is tossed six times in a row and four different such trials are conducted. One trial implies six tosses of the coin. If H stands for head and T stands for tail, the following are the observations from the four trials:

(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.

Which statement describing the last two coin tosses of the fourth trial has the highest probability of being correct?

The correct option is "One H and one T will occur."

Step-by-step Explanation:

1. Understanding the Independence of Coin Tosses:
A coin toss is an independent event. The outcome of any toss does not depend on the results of the previous tosses. Therefore, even though we are observed to have three heads in the first four tosses of the fourth trial (HHHT), the probability of getting a Head (H) or a Tail (T) on the fifth and sixth tosses remains unaffected by the prior tosses.

For an unbiased coin, the probability of obtaining a Head (H) on any single toss is:
$$P\left(H\right)=\frac{1}{2}$$
Similarly, the probability of obtaining a Tail (T) on any single toss is:
$$P\left(T\right)=\frac{1}{2}$$

2. Sample Space for the Last Two Tosses:
There are two tosses remaining (the fifth and sixth tosses). The possible outcomes for these two tosses can be represented as:
$$S=\{HH,HT,TH,TT\}$$
Since each toss is independent, the probability of each individual outcome is:
$$P\left(HH\right)=P\left(HT\right)=P\left(TH\right)=P\left(TT\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$$

3. Evaluating the Probability of Each Statement:

* Statement 1: "Two T will occur."
This corresponds to the single outcome {TT}.
$$P\left(\text{Two T}\right)=P\left(TT\right)=\frac{1}{4}=0.25$$

* Statement 2: "One H and one T will occur."
This statement does not specify the order of occurrence. Therefore, it is satisfied by two outcomes: {HT} or {TH}.
$$P\left(\text{One H and one T}\right)=P\left(HT\right)+P\left(TH\right)=\frac{1}{4}+\frac{1}{4}=\frac{2}{4}=0.50$$

* Statement 3: "Two H will occur."
This corresponds to the single outcome {HH}.
$$P\left(\text{Two H}\right)=P\left(HH\right)=\frac{1}{4}=0.25$$

* Statement 4: "One H will be followed by one T."
This specifies a strict order where H must occur first and T second, corresponding only to the outcome {HT}.
$$P\left(H\text{followed by}T\right)=P\left(HT\right)=\frac{1}{4}=0.25$$

4. Conclusion:
Comparing the probabilities, the statement "One H and one T will occur" has a probability of 0.50, which is higher than the probability of any other option (0.25). Thus, it has the highest probability of being correct.