Consider an experiment of tossing a coin repeatedly until outcomes of two consecutive tosses are sam

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14.Probability

hard

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is.

$\frac{1}{3}$

$\frac{5}{21}$

$\frac{4}{21}$

$\frac{2}{7}$

(IIT-2023)

Solution

$\mathrm{P}(\mathrm{H})=\frac{1}{3} ; \mathrm{P}(\mathrm{T})=\frac{2}{3}$

$\text { Req. prob }=\mathrm{P}(\mathrm{HH} \text { or HTHH or HTHTHH or } \ldots \ldots .)$

$+\mathrm{P}(\mathrm{THH} \text { or THTHH or THTHTHH or } \ldots .)$

$=\frac{\frac{1}{3} \cdot \frac{1}{3}}{1-\frac{2}{3} \cdot \frac{1}{3}}+\frac{\frac{2}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}}{1-\frac{2}{3} \cdot \frac{1}{3}}=\frac{5}{21}$

Standard 11

Mathematics

Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then

hard

(JEE MAIN-2023)

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easy

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medium

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is.

Solution

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