you are dealt 13-card bridge hand from a 52-card bridge deck.

what is the probability that you will be dealt exactly 5 hearts?
what is the probability that you will dealt at list 2 hearts?
what is the probability that you will receive 5 hearts, 4 clubs, 2 diamonds, and 1 spade?

To find the probabilities, you need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

1. Probability of being dealt exactly 5 hearts:
There are 13 hearts in a deck of 52 cards. Choose 5 out of the 13 hearts, and the remaining 8 cards can be any of the other 39 non-heart cards. So, the number of favorable outcomes is (13 choose 5) x (39 choose 8). The total number of possible outcomes is (52 choose 13) since you are dealt 13 cards from a deck of 52 cards. Therefore, the probability is: (13 choose 5) x (39 choose 8) / (52 choose 13).

2. Probability of being dealt at least 2 hearts:
To calculate this probability, you need to consider three cases: receiving exactly 2 hearts, exactly 3 hearts, and so on until receiving all 5 hearts. Calculate the probability for each case and sum them up.

- Probability of receiving exactly 2 hearts:
Choose 2 out of the 13 hearts, and the remaining 11 cards can be any of the other 39 non-heart cards. So, the number of favorable outcomes is (13 choose 2) x (39 choose 11).

- Probability of receiving exactly 3 hearts:
Choose 3 out of the 13 hearts, and the remaining 10 cards can be any of the other 39 non-heart cards. So, the number of favorable outcomes is (13 choose 3) x (39 choose 10).

- Probability of receiving exactly 4 hearts:
Choose 4 out of the 13 hearts, and the remaining 9 cards can be any of the other 39 non-heart cards. So, the number of favorable outcomes is (13 choose 4) x (39 choose 9).

- Probability of receiving exactly 5 hearts (calculated in the previous question).

Sum up the number of favorable outcomes for each case and divide by the total number of possible outcomes, which is (52 choose 13).

3. Probability of receiving 5 hearts, 4 clubs, 2 diamonds, and 1 spade:
To calculate this probability, you need to consider the specific combination of cards described. Choose 5 out of the 13 hearts, 4 out of the 13 clubs, 2 out of the 13 diamonds, and 1 out of the 13 spades. The remaining 1 card can be any of the other 39 non-heart, non-club, non-diamond, non-spade cards. So, the number of favorable outcomes is (13 choose 5) x (13 choose 4) x (13 choose 2) x (13 choose 1) x (39 choose 1). Divide this by the total number of possible outcomes, which is (52 choose 13).

To calculate the probability of being dealt certain cards in a bridge hand, we need to use the concept of combinations.

There are a total of C(52,13) ways to choose 13 cards from a standard deck of 52 cards. This is equal to the number of possible bridge hands.

1. Probability of being dealt exactly 5 hearts:
To calculate this, we need to determine the number of hands that contain exactly 5 hearts, and then divide it by the total number of possible hands.

The number of hands with exactly 5 hearts can be determined by selecting 5 hearts from the total of C(13,5), and the remaining 8 cards can be any non-heart cards selected from C(39,8).

So, the probability of being dealt exactly 5 hearts is:
P(5 hearts) = (C(13,5) * C(39,8)) / C(52,13)

2. Probability of being dealt at least 2 hearts:
To calculate this, we need to determine the number of hands that contain at least 2 hearts, which includes hands with 2, 3, 4, or 5 hearts.

The number of hands can be determined by summing up the number of hands with 2 hearts, 3 hearts, 4 hearts, and 5 hearts.

P(at least 2 hearts) = (C(13,2) * C(39,11) + C(13,3) * C(39,10) + C(13,4) * C(39,9) + C(13,5) * C(39,8)) / C(52,13)

3. Probability of receiving 5 hearts, 4 clubs, 2 diamonds, and 1 spade:
To calculate this, we need to determine the number of hands that contain exactly the specified number of each suit, and then divide it by the total number of possible hands.

The number of hands with the specified suits can be determined by selecting the required number of each suit from the total possible combinations.

P(5 hearts, 4 clubs, 2 diamonds, 1 spade) = (C(13,5) * C(13,4) * C(13,2) * C(13,1)) / C(52,13)

By substituting the values into the formulas, you can calculate the probabilities of each scenario.