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An unbiased coin is tossed until the first head appears or until four tosses are completed, whichever happens earlier. Which of the following statements is/are correct?
$1$ The probability that no head is observed is $\frac{1}{16}$.
$2$ The probability that the experiment ends with three
tosses is $\frac{1}{8}$
Select the correct answer using the code given below:
Options:
$1$ The probability that no head is observed is $\frac{1}{16}$.
$2$ The probability that the experiment ends with three
tosses is $\frac{1}{8}$
Select the correct answer using the code given below:
Solution:
1401 Upvotes
Verified Answer
The correct answer is:
Both 1 and 2
$\mathrm{S}=\{\mathrm{H}, \mathrm{TH}, \mathrm{TTH}$, TTTH, TTTT $\}$
$\mathrm{P}(\mathrm{T})=\mathrm{P}(\mathrm{H})=\frac{1}{2}$
[Probability of getting head or tail in a single toss] $\therefore$ Probability that no head is observed $=\mathrm{P}(\mathrm{TTTT})$ $=\mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T})$
$=\frac{1}{2^{4}}=\frac{1}{16}$
And the probability that the experiment ends with 3 tosses
$=\mathrm{P}(\mathrm{TTH})$
$=\mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{H})$
$=\frac{1}{8}$
Hence, both statements are correct.
$\mathrm{P}(\mathrm{T})=\mathrm{P}(\mathrm{H})=\frac{1}{2}$
[Probability of getting head or tail in a single toss] $\therefore$ Probability that no head is observed $=\mathrm{P}(\mathrm{TTTT})$ $=\mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T})$
$=\frac{1}{2^{4}}=\frac{1}{16}$
And the probability that the experiment ends with 3 tosses
$=\mathrm{P}(\mathrm{TTH})$
$=\mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{T}) \mathrm{P}(\mathrm{H})$
$=\frac{1}{8}$
Hence, both statements are correct.
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