Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is \(\frac{1}{3}\), then the probability that the experiment stops with the head is
- \(\frac{1}{3}\)
- \(\frac{5}{21}\)
- \(\frac{4}{21}\)
- \(\frac{2}{7}\)
The Correct Option is B
Approach Solution - 1
The correct option is: (B) \(\frac{5}{21}\)
P(H) = \(\frac{1}{3}\)
P(T) = \(\frac{2}{3}\)
P(E) = P(HH) + P(THH) + P(HTHH) + P(THTHH) + P(HTHTHH) + P(THTHTHH) + .....
= \(\frac{1}{3^2}+\frac{2}{3^3}+\frac{2}{3^4}+\frac{4}{3^5}+\frac{4}{3^6}+\frac{8}{3^7}+\frac{8}{3^8}+.......\)
\(=(\frac{1}{3^2}+\frac{2}{3^4}+\frac{4}{3^6}+.......)+(\frac{2}{3^3}+\frac{4}{3^5}+\frac{8}{3^7}+.......)\)
P(E) = \(\frac{1}{7}+\frac{2}{21}=\frac{5}{21}\)
Approach Solution -2
The probability of getting heads (H) is \(\frac{1}{3}\). Therefore, the probability of getting tails (T) is \(\frac{2}{3}\), since the total probability of getting either heads or tails must sum to 1.
Case 1. It start with a head and end with HH include examples like HH, HTHH, HTHTHH, and so on.
Case 2. It start with a tail and end with HH include examples like THH, THTHH, THTHTHH, and so on.
Case 1.
HH=\([\frac{1}{3}]^2=\frac{1}{9}\), HTHH=\(\frac{1}{3}.\frac{2}{3}.[\frac{1}{3}]^2=\frac{2}{81}\) and so on
The sum of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\), where aaa is the first term of the series and r is the common ratio.
\(r=P(HT)=\frac{1}{3}.\frac{2}{3}=\frac{2}{9}\), \(a=\frac{1}{9}\)
\(S_1=\frac{\frac{1}{9}}{1-\frac{2}{9}}=\frac{\frac{1}{9}}{\frac{7}{9}}=\frac{1}{7}\)
Case 2.
THH=\(\frac{2}{3}.[\frac{1}{3}]^2=\frac{2}{27}\), THTHH=\(\frac{2}{3}.\frac{1}{3}.\frac{2}{3}.[\frac{1}{3}]^2=\frac{2}{243}\)
\(r=P(TH)=\frac{2}{9}\), \(a=\frac{2}{27}\)
\(S_2=\frac{\frac{2}{27}}{1-\frac{2}{9}}=\frac{\frac{2}{27}}{\frac{7}{9}}=\frac{2}{21}\)
Now, to calculate the total probability
\(P(HH)=S_1+S_2=\frac{1}{7}+\frac{2}{21}=\frac{5}{21}\)
So, the correct option is (B): \(\frac{5}{21}\)
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.