Select correct alternative from the given choices. A and B...

Select correct alternative from the given choices. A and $B$ pick a card at random from a well shuffled pack of cards, one after the other replacing it every time till one of them gets a spade. The person who picks a spade is declared the winner. If A begins the game, then the probability that $\mathrm{B}$ wins the game is (A) $\frac{5}{9}$ (B) $\frac{4}{9}$ (C) $\frac{3}{7}$ (D) $\frac{4}{7}$

Select correct alternative from the given choices.
A and $B$ pick a card at random from a well shuffled pack of cards, one after the other replacing it every time till one of them gets a spade. The person who picks a spade is declared the winner. If A begins the game, then the probability that $\mathrm{B}$ wins the game is
(A) $\frac{5}{9}$
(B) $\frac{4}{9}$
(C) $\frac{3}{7}$
(D) $\frac{4}{7}$

Key Concepts

Replacement in Probability

In probability theory, sampling with replacement means that after each event the object (in this case, a card) is returned to the population, keeping the total number and composition of objects constant for each trial. This ensures that the probability of a particular outcome remains the same on every draw, which is essential for modeling repeated independent trials.

Independent Trials

Because the card is replaced after each draw, each trial is independent of the previous ones. This independence implies that the outcome of one trial does not affect the outcome of another, allowing the use of methods that assume constant probabilities throughout the experiment.

Geometric Series in Repeated Trials

When an experiment is repeated until a success is achieved, such as drawing a particular type of card, the series of probabilities forms a geometric series. This series represents the sum of probabilities over an indefinite number of trials, where each term is the probability of failing after several attempts followed by a success, and it sums up to the total probability of eventually achieving the desired outcome.

Recursion in Probabilistic Processes

In processes that repeat under the same conditions, recursion is used to set up equations that express the overall probability of an event in terms of outcomes in the first trial and the probability of the event occurring in subsequent trials. This approach provides a systematic method to derive the probability of a win or success by considering both immediate and repeated future attempts under identical conditions.

Transcript

00:01 Hello students, our question is a and b pick a cart, a random formable shuffle, pegs of cards.

00:11 One after the other, replacing it every time, till one of them gets a spake.

00:20 The person who picks a speck is declared the winner.

00:25 If a begins the game, then the probability that b wins the game is.

00:32 So here a and b are picking cards from well shuffle take of cards.

00:40 So we have to find the winning probability of b when b gets a spake.

00:50 He will be a big way.

00:52 So let's start our solution.

00:54 We have to find the probability of b wins.

01:01 B wins.

01:04 So as we know that there are total 15 ,000.

01:09 Cards in the pay total cards is equal to total cards is equal to 50 so there are total state total spade is equal to equal to 30 so probability of probability of a spare that is getting a spade is equal to 13 out of 52 which is equal to 13 out of which is equal to one out of four and probability of not getting a spade that is not getting spared not getting a spade is equal to one minus one out of four which is equal to three out of now since a starts again since a starts the game then a starts again then a starts again so the first pick will be done by a, that is, first pick and second pick will be done by b, second pick will be done by b, then third pick will be done by again a, fourth pick will be done by b, fifth pick will be done by a, six pick will be done by d, and this will continue on, so on...

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