If a person draws two cards from a standard deck (without replacing them), what is the probability that at least one of the cards is a face card? (Round your answer to one decimal place.)
Oh, probabilities, the clowns of mathematics! Let's crunch some numbers, shall we? There are a total of 52 cards in a standard deck, and out of those, 12 are face cards. Now, when you draw the first card, you have a 12/52 chance of drawing a face card. But remember, you don't replace the first card, so for the second card there are only 51 cards left and 11 face cards.
Now, we need to calculate the probability of drawing no face cards at all. The chance of drawing a non-face card on the first draw is 40/52, and on the second draw it's 39/51. We multiply these probabilities together to find the probability of drawing no face cards: (40/52) * (39/51) = 0.588.
Finally, we subtract this probability from 1 to find the probability of drawing at least one face card: 1 - 0.588 = 0.412.
So the probability of at least one face card is approximately 0.4, or 40%. But hey, numbers aren't as fun as juggling, so keep that in mind!
To find the probability that at least one of the cards is a face card, we need to calculate the complement of the event that neither of the cards is a face card.
Step 1: Determine the total number of possible outcomes when drawing two cards from a standard deck without replacement.
In this case, there are 52 cards in a standard deck, so there are a total of 52 ways to choose the first card. Once the first card is chosen, there are 51 remaining cards to choose from for the second card. Therefore, the total number of possible outcomes is 52 * 51 = 2,652.
Step 2: Determine the number of favorable outcomes when neither of the cards is a face card.
A standard deck contains 12 face cards (Jack, Queen, and King of each suit), so there are 40 non-face cards. The number of ways to choose two non-face cards from these 40 cards is given by the combination formula:
C(40, 2) = 40! / (2!(40-2)!) = 780.
Step 3: Calculate the probability that neither of the cards is a face card.
The probability of an event occurring is the number of favorable outcomes divided by the number of possible outcomes.
P(Neither of the cards is a face card) = 780 / 2,652 ≈ 0.2941
Step 4: Calculate the probability that at least one of the cards is a face card.
The probability of an event occurring is equal to 1 minus the probability of its complement.
P(At least one of the cards is a face card) = 1 - P(Neither of the cards is a face card)
P(At least one of the cards is a face card) = 1 - 0.2941 ≈ 0.7059
Therefore, the probability that at least one of the cards is a face card is approximately 0.7059 or 70.6% (rounded to one decimal place).
To calculate the probability of at least one card being a face card when two cards are drawn from a standard deck, we need to determine the total number of favorable outcomes and the total number of possible outcomes.
Step 1: Determine the total number of possible outcomes.
In a standard deck of 52 cards, there are 4 face cards in each suit (Jack, Queen, King), making a total of 12 face cards. Since we are not replacing the cards after drawing, the total number of possible outcomes for the first card drawn is 52. After drawing the first card, there are 51 cards left for the second draw.
Step 2: Determine the total number of favorable outcomes.
To determine the number of favorable outcomes, we need to consider two scenarios:
1. Drawing at least one face card in the first draw and any card in the second draw.
2. Drawing a non-face card in the first draw and a face card in the second draw.
Scenario 1: Drawing at least one face card in the first draw and any card in the second draw.
The number of favorable outcomes for this scenario is calculated as follows:
Number of ways to draw at least one face card in the first draw: 4 face cards out of 52 (total number of cards).
Number of ways to draw any card in the second draw: 52 (since any card can be drawn, regardless of the first draw).
So, the total number of favorable outcomes for this scenario is 4 * 52.
Scenario 2: Drawing a non-face card in the first draw and a face card in the second draw.
The number of favorable outcomes for this scenario is calculated as follows:
Number of ways to draw a non-face card in the first draw: 40 non-face cards out of 52.
Number of ways to draw a face card in the second draw: 12 face cards out of the remaining 51 cards after the first draw.
So, the total number of favorable outcomes for this scenario is 40 * 12.
Step 3: Calculate the probability.
To calculate the probability, we divide the total number of favorable outcomes by the total number of possible outcomes:
Total number of favorable outcomes = (4 * 52) + (40 * 12)
Total number of possible outcomes = 52 * 51
Finally, we calculate the probability and round it to one decimal place:
Probability = (Total number of favorable outcomes / Total number of possible outcomes) * 100
= (4 * 52 + 40 * 12) / (52 * 51) * 100
Performing the calculations:
Probability = 434 / 2652 * 100
≈ 16.35% (rounded to one decimal place)
Therefore, the probability that at least one of the cards drawn is a face card is approximately 16.4%.
This is the same as finding one or two face cards.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Two face cards = 12/52 * 11/51 = ?
One face card = 12/52 * (51-11)/51 = ?
Either-or probabilities are found by adding the individual probabilities.