Consider the experiment of choosing the top 7 cards from a shuffled deck in two ways:

(a) Without replacement.
1. What is the sample space?
2. What is the probability that you get only red cards?
(b) With replacement.
1. What is the sample space? What is the probability that you get no spades?

To answer these questions, let's break them down one by one.

(a) Without replacement:

1. What is the sample space?
The sample space refers to the set of all possible outcomes. In this case, we are choosing 7 cards from a shuffled deck without replacement. The sample space is the number of ways we can choose 7 cards from the deck without considering the order in which they are chosen.

To calculate the sample space, we need to use the concept of combinations. The combination formula is given by nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen.

In this case, the deck has 52 cards and we want to choose 7 cards. So the sample space can be calculated as 52C7 = 52! / (7!(52-7)!) = 133,784,560.

2. What is the probability that you get only red cards?
To calculate the probability of getting only red cards, we need to determine the number of favorable outcomes (getting only red cards) and divide it by the total number of possible outcomes (sample space).

In a standard deck of 52 cards, half of the cards (26) are red. Therefore, the probability of drawing a red card on the first draw is 26/52. After the first draw, there are 25 red cards remaining out of a total of 51 cards. So the probability of drawing a red card on the second draw is 25/51. This process continues for each subsequent draw.

Since we want to choose 7 cards, we multiply the individual probabilities for each draw. Hence, the probability of getting only red cards can be calculated as (26/52) * (25/51) * (24/50) * (23/49) * (22/48) * (21/47) * (20/46) = approximately 0.00264115.

(b) With replacement:

1. What is the sample space? What is the probability that you get no spades?
When choosing cards with replacement, after each draw, the chosen card is put back into the deck. This means that the total number of cards remains the same for each draw, and every card has an equal probability of being selected on every draw.

Therefore, the sample space for choosing 7 cards with replacement is simply 52^7 since we have 52 options for each of the 7 cards.

To calculate the probability of getting no spades, we need to determine the number of favorable outcomes (getting no spades) and divide it by the total number of possible outcomes (sample space).

In a standard deck of 52 cards, there are 13 spades. Since each draw is independent, the probability of not drawing a spade on any single draw is (52-13)/52 = 39/52. As we are choosing 7 cards, this probability needs to be multiplied 7 times.

Hence, the probability of drawing no spades when choosing 7 cards with replacement is (39/52)^7, which is approximately 0.049.