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?
Two fair dice are thrown. The number of cases where the number appearing on the upper face of the first die is not less than that on the lower face of the second die is
The number of Hens, Ducks, and Goats in farm P are 65, 91 and 169, respectively. The total number of Hens, Ducks and Goats in a nearby farm Q is 416. The ratio of hens : ducks : goats in farm Q is 5 : 14 : 13. All the hens, ducks and goats are sent from farm Q to farm P.
The new ratio of hens : ducks : goats in farm P is ____________.
A person divided an amount of Rs. 100,000 into two parts and invested in two different schemes. In one he got 10% profit and in the other he got 12%. If the profit percentages are interchanged with these investments he would have got Rs.120 less. Find the ratio between his investments in the two schemes.
S, M, E and F are working in shifts in a team to finish a project. M works with twice the efficiency of others but for half as many days as E worked. S and M have 6 hour shifts in a day, whereas E and F have 12 hours shifts. What is the ratio of contribution of M to contribution of E in the project?
