From a deck of 52 cards, 7 are drawn randomly without replacement. Let X and Y be the number of hearts and spades respectively .

a) What is P(X > or = Y)?

To find P(X ≥ Y), we need to calculate the probability that the number of hearts (X) is greater than or equal to the number of spades (Y).

To begin, let's calculate the total number of possible outcomes of drawing 7 cards from a deck of 52 cards without replacement. This can be evaluated using the concept of combinations. The number of combinations or ways to choose 7 cards from a deck of 52 is given by the formula C(52, 7), which is evaluated as:

C(52, 7) = 52! / (7! * (52-7)!) = 133,784,560

Now, let's determine the number of favorable outcomes, which is the number of ways we can draw 7 cards such that the number of hearts (X) is greater than or equal to the number of spades (Y).

When X > Y:
- We can select the hearts in various ways, starting from 1 heart up to 7 hearts.
- For each number of hearts, the remaining cards will be spades.
- The remaining number of cards can range from 0 (all hearts) to 7 (no hearts).
- So, the favorable outcomes can be calculated as the sum of the number of ways to choose hearts and corresponding spades for each case.

The calculation for each case is:
- For X = 1: Choose 1 heart and 6 spades --> C(13, 1) * C(39, 6)
- For X = 2: Choose 2 hearts and 5 spades --> C(13, 2) * C(39, 5)
- For X = 3: Choose 3 hearts and 4 spades --> C(13, 3) * C(39, 4)
- For X = 4: Choose 4 hearts and 3 spades --> C(13, 4) * C(39, 3)
- For X = 5: Choose 5 hearts and 2 spades --> C(13, 5) * C(39, 2)
- For X = 6: Choose 6 hearts and 1 spade --> C(13, 6) * C(39, 1)
- For X = 7: Choose all 7 hearts and no spades --> C(13, 7) * C(39, 0)

Adding up these possibilities, we get the total number of favorable outcomes.

Finally, divide the number of favorable outcomes by the total number of possible outcomes to find the probability:

P(X ≥ Y) = (sum of favorable outcomes) / (total number of possible outcomes)