Determine the odds of choosing 4 aces when drawing 4 cards randomly from a regular deck of cards.
I tried to use the formula p(a)/p(a') but I don't know how to find the p(a) and p(a') values^
p(1st ace)=4/52
p(2nd ace)=3/51
...
p(all 4 aces)= (4*3*2*1)/(52*51*50*49) = 4!/(52P4) = 1/(52C4) = 1/270725
so, the odds are 1:270724
Ohh this is a B given A question that includes permutations. I didn't see that. Thank you!
To determine the odds of choosing 4 aces when drawing 4 cards randomly from a regular deck of cards, we need to calculate the probability of this specific event occurring.
First, let's find the total number of ways to choose 4 cards from a deck of 52 cards. This can be calculated using the combination formula, which is represented as "nCr". In this case, we have 52 cards to choose from, and we want to choose 4 cards.
The formula for nCr is:
nCr = n! / (r! * (n-r)!)
Where:
n = total number of items to choose from (52 cards in this case)
r = number of items to choose (4 cards in this case)
! = factorial
So, applying the formula:
nCr = 52! / (4! * (52-4)!)
Simplifying the formula:
52! = 52 * 51 * 50 * 49 * 48!
After canceling out the factorials:
nCr = (52 * 51 * 50 * 49 * 48!) / (4 * 3 * 2 * 1 * 48!)
The 48! terms cancel out:
nCr = (52 * 51 * 50 * 49) / (4 * 3 * 2 * 1)
nCr = 270,725
Therefore, there are 270,725 different ways to choose 4 cards from a deck of 52 cards.
Next, we need to determine the number of ways to choose 4 aces from the deck. Since there are 4 aces in a deck, the number of ways to choose 4 aces can be calculated as:
nCr_aces = 4! / (4! * (4-4)!)
Simplifying the equation:
nCr_aces = 4! / (4! * 0!)
Since 0! is defined as 1:
nCr_aces = 4! / 4!
nCr_aces = 1
Therefore, there is only 1 way to choose 4 aces from the deck.
To find the probability, we divide the number of favorable outcomes (choosing 4 aces) by the number of possible outcomes (choosing any 4 cards):
Probability = nCr_aces / nCr
Probability = 1 / 270,725
Finally, to express the odds, we can use the ratio of favorable outcomes to unfavorable outcomes:
Odds = 1 : (270,725 - 1)
Hence, the odds of choosing 4 aces when drawing 4 cards randomly from a regular deck of cards is 1 in 270,724.