A person draws a card from a pack of 52 playing cards,...

A person draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues doing this until be draws a spade, the chance that he will fail in the first two draws is
(A) $\frac{1}{16}$
(B) $\frac{9}{16}$
(C) $\frac{9}{64}$
(D) $\frac{1}{64}$

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Key Concepts

Multiplication Rule of Probability

The multiplication rule of probability states that for a sequence of independent events, the probability of all events occurring is the product of the individual probabilities. This rule is essential in calculating the likelihood of consecutive failures or successes in repeated experiments.

Event Complement

The event complement refers to calculating the probability of an event not occurring. In many probability problems, finding the probability of failure (or non-occurrence) is achieved by considering all associated outcomes that do not meet the success criteria.

Sampling with Replacement

Sampling with replacement occurs when an item is returned to the population after each trial. This process ensures that the probability distribution remains constant throughout the series of experiments, which is crucial when calculating repeated probabilities.

Independent Events

Independent events are those whose outcomes do not affect one another. When an experiment is repeated, such as drawing a card with replacement, the probability of any outcome remains unchanged on each trial, making each draw independent.

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