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Sebuah kartu remi diacak tak beraturan, dan diambil sebanyak $5$ kartu berurutan. Berapa probabilitas terambil $3$ kartu berwarna hitam secara berurutan?

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Convert, compare, add, and subtract mixed customary units

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Q. Sebuah kartu remi diacak tak beraturan, dan diambil sebanyak

5

kartu berurutan. Berapa probabilitas terambil

3

kartu berwarna hitam secara berurutan?

Understand Deck Composition: First, we need to understand the composition of a deck of cards. A standard deck has $52$ cards, with $26$ red cards (hearts and diamonds) and $26$ black cards (spades and clubs). We want to calculate the probability of drawing $3$ consecutive black cards in a sequence of $5$ cards.
Calculate Probability of $3$ Black Cards: To find the probability of drawing $3$ consecutive black cards, we need to consider the different ways $3$ black cards can appear in a sequence of $5$ cards. The $3$ black cards can be in the following positions: $(1, 2, 3)$ , $(2, 3, 4)$ , or $(3, 4, 5)$ .
Consider Different Positions: For each of these positions, we calculate the probability separately and then add them up, since these are mutually exclusive events (they cannot happen at the same time). $\newline$ Let's calculate the probability for the first position $(1, 2, 3)$ : $\newline$ The probability of drawing the first black card is $\frac{26}{52}$ . $\newline$ The probability of drawing the second black card is $\frac{25}{51}$ (since one black card is already drawn). $\newline$ The probability of drawing the third black card is $\frac{24}{50}$ . $\newline$ The probability of the fourth and fifth cards being non-black is $\frac{26}{49}$ and $\frac{25}{48}$ , respectively.
Calculate Probability for First Position: Now, we multiply these probabilities to find the probability for the first position $(1, 2, 3)$ : $P(1, 2, 3) = \left(\frac{26}{52}\right) * \left(\frac{25}{51}\right) * \left(\frac{24}{50}\right) * \left(\frac{26}{49}\right) * \left(\frac{25}{48}\right)$

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