One card is drawn at random from a deck of 52 cards. The first card is not replaced, and a second card is drawn. Find the probability that both cards are clubs.
The correct answer and explanation is :
To determine the probability of drawing two clubs consecutively from a standard 52-card deck without replacement, we can follow these steps:
- First Draw: The probability of drawing a club on the first draw is calculated by dividing the number of clubs by the total number of cards: [ P(\text{First Club}) = \frac{13}{52} = \frac{1}{4} ]
- Second Draw: After drawing one club, there are now 12 clubs left in a deck of 51 remaining cards. The probability of drawing a club on the second draw is: [ P(\text{Second Club | First Club}) = \frac{12}{51} ]
- Combined Probability: The probability of both events occurring (drawing two clubs in succession without replacement) is found by multiplying the probabilities of each individual event: [ P(\text{Two Clubs}) = P(\text{First Club}) \times P(\text{Second Club | First Club}) = \frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} ]
- Simplification: Simplifying the fraction: [ \frac{156}{2652} = \frac{1}{17} ]
Therefore, the probability of drawing two clubs consecutively without replacement is ( \frac{1}{17} ), or approximately 0.0588 (5.88%).
Explanation:
In a standard deck, there are 52 cards divided into four suits: clubs, diamonds, hearts, and spades, each containing 13 cards. When calculating probabilities without replacement, it’s essential to adjust the total number of remaining cards after each draw.
- First Draw: The chance of drawing a club is straightforward: 13 clubs out of 52 total cards, yielding a probability of ( \frac{13}{52} ).
- Second Draw: Once a club has been drawn, only 12 clubs remain, and the deck now contains 51 cards. Thus, the probability of drawing another club is ( \frac{12}{51} ).
Multiplying these probabilities gives the combined probability of both events occurring in sequence.
This method aligns with the multiplication rule in probability, which states that the probability of two independent events both occurring is the product of their individual probabilities. However, in this scenario, the events are not entirely independent because the outcome of the first draw influences the second. This dependency is accounted for by adjusting the probabilities accordingly after the first draw.
Understanding this concept is crucial in scenarios where selections are made without replacement, as each selection alters the composition of the sample space, thereby affecting subsequent probabilities.