Alice plays the following game with Bob. First, Alice randomly chooses a set of 4 cards out of a 52-card deck, memorizes them, and places them back into the deck. (Any set of 4 cards is equally likely.) Then, Bob randomly chooses 8 cards out of the same deck. (Any set of 8 cards is equally likely.)
What is the probability that all 4 cards Alice chose were also among the 8 cards chosen by Bob?
(48)
(4)
-----
(52)
(8)
To solve this problem, we can use the concept of combinations and the principle of conditional probability.
Step-by-step solution:
Step 1: Calculate the total number of possible sets of 8 cards that Bob can choose from the 52-card deck. We can use the combination formula to calculate this. The formula for combinations is given by nCr, where n is the total number of items to choose from and r is the number of items to choose. In this case, we have 52 cards to choose from and Bob chooses 8 cards, so the total number of possible sets of 8 cards is given by 52C8.
52C8 = (52!)/(8! * (52-8)!)
= (52!)/(8! * 44!)
= (52 * 51 * 50 * 49 * 48 * 47 * 46 * 45)/(8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
≈ 63,063,000
So, there are approximately 63,063,000 possible sets of 8 cards that Bob can choose from.
Step 2: Calculate the total number of possible sets of 8 cards that include the 4 cards chosen by Alice. Since Alice chooses 4 cards out of the 52-card deck, the remaining 48 cards are available for Bob to choose from. We can calculate the total number of possible sets of 8 cards that include the 4 cards chosen by Alice using the combination formula. The formula for combinations is given by nCr, where n is the total number of items to choose from and r is the number of items to choose. In this case, we have 48 cards to choose from (after Alice removes her 4 cards) and Bob chooses 8 cards, so the total number of possible sets of 8 cards that include the 4 cards chosen by Alice is given by 48C4.
48C4 = (48!)/(4! * (48-4)!)
= (48!)/(4! * 44!)
= (48 * 47 * 46 * 45)/(4 * 3 * 2 * 1)
≈ 194,580
So, there are approximately 194,580 possible sets of 8 cards that include the 4 cards chosen by Alice.
Step 3: Calculate the probability that all 4 cards chosen by Alice are also among the 8 cards chosen by Bob. To do this, we divide the number of favorable outcomes (i.e., sets of 8 cards that include the 4 cards chosen by Alice) by the total number of possible outcomes (i.e., sets of 8 cards that Bob can choose from). Therefore, the probability is given by:
Probability = (Number of favorable outcomes)/(Total number of possible outcomes)
= 194,580/63,063,000
≈ 0.003084
So, the probability that all 4 cards chosen by Alice are also among the 8 cards chosen by Bob is approximately 0.003084, or about 0.3084%.
To find the probability that all 4 cards Alice chose were also among the 8 cards chosen by Bob, we need to determine two things:
1. The total number of possible sets of 4 cards that Alice can choose from a 52-card deck.
2. The total number of possible sets of 8 cards that Bob can choose from the same deck.
Let's tackle each of these steps:
Step 1: Calculating the number of possible sets of 4 cards chosen by Alice.
To calculate this, we use the concept of combinations. We want to choose 4 cards out of a total of 52 cards, so the number of possible sets of 4 cards can be calculated using the formula for combinations, which is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items to choose from (in this case, 52 cards) and r is the number of items to choose (in this case, 4 cards).
So, the number of possible sets of 4 cards chosen by Alice is:
C(52, 4) = 52! / (4!(52-4)!).
Now, we can simplify this expression:
C(52, 4) = (52 x 51 x 50 x 49) / (4 x 3 x 2 x 1).
Simplifying further, we get:
C(52, 4) = 270,725.
Therefore, there are 270,725 possible sets of 4 cards that Alice can choose.
Step 2: Calculating the number of possible sets of 8 cards chosen by Bob.
Using the same logic as before, we apply the combinations formula:
C(52, 8) = 52! / (8!(52-8)!).
Simplifying this expression gives us:
C(52, 8) = 752,538,150.
Therefore, there are 752,538,150 possible sets of 8 cards that Bob can choose.
Finally, to find the probability that all 4 cards Alice chose were also among the 8 cards chosen by Bob, we divide the number of favorable outcomes (which is 270,725) by the number of possible outcomes (which is 752,538,150).
Probability = (Number of favorable outcomes) / (Number of possible outcomes)
= 270,725 / 752,538,150
Calculating this division gives us the probability:
Probability ≈ 0.0003597, or approximately 0.036%.
Therefore, the probability that all 4 cards Alice chose were also among the 8 cards chosen by Bob is approximately 0.036%.