What is the probability that a card drawn randomly from a standard deck of 52 cards is a black seven? Express your answer as a fraction in lowest terms or a decimal number rounded to three decimal places, if necessary.
Answer (1)
There are 2 black sevens in a standard 52-card deck.
The probability of drawing a black seven is the number of black sevens divided by the total number of cards: 52 2 .
Simplify the fraction: 52 2 = 26 1 .
Convert to decimal and round to three decimal places: 26 1 ≈ 0.038 .
26 1
Explanation
Understand the problem and provided data A standard deck of 52 cards contains four suits: hearts, diamonds, clubs, and spades. Hearts and diamonds are red, while clubs and spades are black. Each suit has one card of each rank, including a seven. Therefore, there is one seven of clubs and one seven of spades in a standard deck.
Determine the number of black sevens and total cards We want to find the probability of drawing a black seven from a standard deck of 52 cards. Since there are two black sevens (the seven of clubs and the seven of spades) in the deck, the number of favorable outcomes is 2. The total number of possible outcomes is the total number of cards in the deck, which is 52.
Calculate the probability as a fraction The probability of drawing a black seven is the ratio of the number of black sevens to the total number of cards in the deck. This can be expressed as a fraction: 52 2 .
Simplify the fraction To express the probability in lowest terms, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: 52 ÷ 2 2 ÷ 2 = 26 1 .
Convert to decimal To express the probability as a decimal rounded to three decimal places, we divide 1 by 26: 26 1 ≈ 0.0384615 . Rounding to three decimal places, we get 0.038.
State the final answer The probability of drawing a black seven from a standard deck of 52 cards is 26 1 or approximately 0.038.
Examples
Consider a game where you win if you draw a black seven from a deck of cards. Knowing the probability helps you understand your chances of winning. For example, if you play this game many times, you would expect to win approximately 3.8% of the time. This concept is also useful in casinos and other games of chance to determine the odds and payouts.
