Two cards are drawn at random, one after the other, without replaceme…

The correct answer is Option (4) → \(\displaystyle\frac{25}{102}\)

Let R be the event “the first card is red” and H the event “the second card is a heart”. A red first card is a heart or a diamond, and the number of hearts left depends on which.

First card a heart, then a heart (multiplication theorem):

    \( \displaystyle \frac{13}{52}\times\frac{12}{51} = \frac{156}{2652} \)

First card a diamond, then a heart:

    \( \displaystyle \frac{13}{52}\times\frac{13}{51} = \frac{169}{2652} \)

Adding the two cases:

    \( \displaystyle P(R\cap H) = \frac{325}{2652} \)

Also \(P(R) = \frac{26}{52} = \frac{1}{2}\). By the definition of conditional probability:

    \( \displaystyle P(H|R) = \frac{325}{2652}\div\frac{1}{2} \)

    \( \displaystyle = \frac{650}{2652} = \frac{25}{102} \)

Why the other options are incorrect

• Option (1) \(\displaystyle\frac{13}{51}\) — Incorrect. This is the answer only when the first card is known to be a diamond.

• Option (2) \(\displaystyle\frac{4}{17}\) — Incorrect. This is \(\frac{12}{51}\), the answer only when the first card is known to be a heart.

• Option (3) \(\displaystyle\frac{1}{4}\) — Incorrect. This is the chance that the second card is a heart when nothing is known about the first card.