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?
One H and one T will occur.
Understanding the probabilities of an unbiased coin toss is fundamental in probability theory. The question asks us to determine the most probable outcome for the last two tosses of the fourth trial (HHHT__ __) of an unbiased coin.
A key concept for solving this problem is that each coin toss is an independent event. This means the outcome of previous tosses does not influence the outcome of any future tosses. For an unbiased coin, the probability of getting a Head (H) is equal to the probability of getting a Tail (T).
Coin Toss Probabilities Defined
For an unbiased coin, the probability of each outcome is:
- Probability of getting a Head (H), denoted as $P(H) = \frac{1}{2}$
- Probability of getting a Tail (T), denoted as $P(T) = \frac{1}{2}$
Since each toss is independent, the probability of a sequence of tosses is the product of the probabilities of individual tosses.
Analyzing the Fourth Trial
The fourth trial observation is HHHT__ __. We need to predict the last two outcomes. Let's consider all possible outcomes for these two remaining tosses and calculate their probabilities.
Possible Outcomes for Two Coin Tosses
When tossing a coin two times, there are four equally likely possible outcomes:
- HH (Head, Head): Both tosses result in a Head.
- HT (Head, Tail): The first toss is a Head, and the second is a Tail.
- TH (Tail, Head): The first toss is a Tail, and the second is a Head.
- TT (Tail, Tail): Both tosses result in a Tail.
Calculating Probabilities for Each Outcome
Let's calculate the probability for each of these two-toss sequences:
- Probability of HH: $P(HH) = P(H) \times P(H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of HT: $P(HT) = P(H) \times P(T) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of TH: $P(TH) = P(T) \times P(H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
- Probability of TT: $P(TT) = P(T) \times P(T) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
Evaluating the Options for Highest Probability
Now, let's look at each statement provided in the options and calculate its probability based on our findings:
| Statement Description | Corresponding Outcomes | Probability Calculation | Total Probability |
|---|---|---|---|
| Two T will occur. | TT | $P(TT) = \frac{1}{4}$ | $\frac{1}{4}$ |
| One H and one T will occur. | HT or TH | $P(HT) + P(TH) = \frac{1}{4} + \frac{1}{4}$ | $\frac{2}{4} = \frac{1}{2}$ |
| Two H will occur. | HH | $P(HH) = \frac{1}{4}$ | $\frac{1}{4}$ |
| One H will be followed by one T. | HT | $P(HT) = \frac{1}{4}$ | $\frac{1}{4}$ |
Probability Comparison and Conclusion
Comparing the probabilities for each statement:
- Probability of "Two T will occur" is $\frac{1}{4}$.
- Probability of "One H and one T will occur" is $\frac{1}{2}$.
- Probability of "Two H will occur" is $\frac{1}{4}$.
- Probability of "One H will be followed by one T" is $\frac{1}{4}$.
The highest probability among these is $\frac{1}{2}$, which corresponds to the statement "One H and one T will occur." This statement is true if the outcome is HT or TH, making it the most likely scenario for the last two tosses of the unbiased coin.
Important Questions from Numerical Estimation
The number of digits you have to type to write all the page numbers of a book starting from I (first page) is 2019. What is the number of pages in that book?
1200 men and 500 women can build a bridge in 2 weeks. 900 men and 250 women will take 3 weeks to build the same bridge. How many men will be needed to build the bridge in one week?
The number of 3-digit numbers such that the digit 1 is never to the immediate right of 2 is
Given \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\). Which one of the following statements is TRUE?
Two and a quarter hours back, when seen in a mirror, the reflection of a wall clock without number markings seemed to show 1:30. What is the actual current time shown by the clock?
