Two cards are drawn from a well-shuffled deck of 52 playing cards. Let X denote the number of aces drawn. Find P(X = 0). (Round your answer to three decimal places.)
p none on first draw = 48/52
now I have 4 aces in 51 cards
p none on second draw = 47/51
so 48/52 * 47/51 = .851
Well, the chances of drawing an ace on the first draw is 4 out of 52 since there are 4 aces in a deck of 52 cards.
After the first card is drawn, there are now 51 cards left in the deck, and only 3 aces left. So the chances of not drawing an ace on the second draw is 48 out of 51.
To find the probability of not drawing any aces at all, you just multiply these probabilities together:
P(X = 0) = (48/51) * (4/52)
Now let me fetch my trusty calculator...
Calculating...
P(X = 0) ≈ 0.694
So, the probability of not drawing any aces is approximately 0.694, or in other words, a slim chance. Hopefully, lady luck will be on your side next time!
To find the probability of drawing 0 aces, we need to calculate the ratio of the number of favorable outcomes (drawing 0 aces) to the number of possible outcomes.
Number of favorable outcomes: To draw 0 aces, both cards drawn must not be an ace. There are 48 non-ace cards in a deck of 52 cards. So the number of favorable outcomes is obtained by selecting 2 non-ace cards out of the 48: C(48, 2) = 48! / (2!(48-2)!) = 48! / (2! x 46!) = (48 x 47) / (2 x 1) = 2256.
Number of possible outcomes: The number of ways to draw 2 cards from a deck of 52 cards is given by the combination notation C(52, 2) = 52! / (2!(52-2)!) = 52! / (2! x 50!) = (52 x 51) / (2 x 1) = 1326.
Now, we can calculate the probability using the formula: P(X = 0) = (Number of favorable outcomes) / (Number of possible outcomes) = 2256 / 1326 ≈ 1.700.
Therefore, the probability of drawing 0 aces is approximately 0.170.
To find the probability of drawing zero aces, we need to determine the number of favorable outcomes (drawing zero aces) and the total number of possible outcomes.
First, let's consider the number of favorable outcomes. We want to draw zero aces, which means we cannot draw any of the four aces from the deck of 52 playing cards. We can calculate this by selecting two cards without any aces, which can be done in the following way:
- Selecting a non-ace card from the 48 remaining non-ace cards: This can be done in C(48, 1) ways.
- Selecting another non-ace card from the remaining 47 non-ace cards: This can be done in C(47, 1) ways.
The number of favorable outcomes is given by the product of these two combinations:
Favorable Outcomes = C(48, 1) * C(47, 1)
Next, let's consider the total number of possible outcomes. We need to choose any two cards from the deck of 52 playing cards, which can be done in C(52, 2) ways.
Total Possible Outcomes = C(52, 2)
Finally, we can calculate the probability P(X = 0) of drawing zero aces by dividing the number of favorable outcomes by the total number of possible outcomes:
P(X = 0) = Favorable Outcomes / Total Possible Outcomes
P(X = 0) = (C(48, 1) * C(47, 1)) / C(52, 2)
Now, let's calculate the values and find the answer:
Favorable Outcomes = C(48, 1) * C(47, 1) = 48 * 47 = 2256
Total Possible Outcomes = C(52, 2) = (52 * 51) / (2 * 1) = 1326
P(X = 0) = 2256 / 1326 ≈ 1.703
Therefore, the probability of drawing zero aces (P(X = 0)) is approximately 0.001.