A single card is selected from an ordinary deck of cards. The sample space is shown in the following figure. Find the following probabilities. (Enter the answers either as fractions or as decimals rounded to three places.)


P(five of clubs) = 1
P(five) = 2
P(club) = 3
P(jack) = 4
P(spade) = 5
P(jack of spades) = 6
P(five and a jack) = 7
P(five or a jack) = 8
P(heart and a jack) = 9
P(heart or a jack) = 10

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Because there's no sample space diagram here to clarify the point, my first question is "Does an ordinary deck of cards contain a joker?", because that affects the answer to every subsequent question. Assuming that it doesn't:

P(five of clubs): there's only one five of clubs in a deck, so the answer is 1/52.
P(five): there are four fives in a deck, so the answer is 4/52 = 1/13.
P(club): there are 13 clubs in a deck, so the answer is 13/52 = 1/4.
And so on. Note on question 7 that a single card can't be both a five and a jack.

To find the probabilities, we need to understand the concepts of probability and how to calculate them.

Probability is a measure of the likelihood of an event occurring. It ranges from 0 (impossible) to 1 (certain). To calculate the probability of an event, we divide the number of favorable outcomes by the total number of possible outcomes.

In this case, we are dealing with a single card selected from an ordinary deck of cards. A standard deck of cards contains 52 cards, divided into 4 suits (hearts, diamonds, clubs, and spades) and 13 ranks (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king). Each suit has 13 cards.

Now let's find the probabilities:

1. P(five of clubs): The five of clubs is only one card out of the 52 cards in the deck. So, the probability is 1/52 or approximately 0.019.

2. P(five): There are four different fives in the deck (one in each suit), so the probability is 4/52 or approximately 0.077.

3. P(club): There are 13 club cards in the deck, so the probability is 13/52 or 1/4, which is also equal to 0.25.

4. P(jack): There are four jacks in the deck (one in each suit), so the probability is 4/52 or approximately 0.077.

5. P(spade): There are 13 spade cards in the deck, so the probability is 13/52 or 1/4, which is also equal to 0.25.

6. P(jack of spades): The jack of spades is only one card out of the 52 cards in the deck. So, the probability is 1/52 or approximately 0.019.

7. P(five and a jack): The five and the jack of any suit are two separate events. The probability of getting a five is 4/52, and the probability of getting a jack is 4/52. Since these events are independent, we can multiply their probabilities: (4/52) * (4/52) = 16/2704 or approximately 0.006.

8. P(five or a jack): To find the probability of the union of two events ("or"), we add their individual probabilities. In this case, we already calculated the probabilities of getting a five and a jack separately, so we add them: (4/52) + (4/52) = 8/52 or 2/13, which is also approximately 0.154.

9. P(heart and a jack): The heart and the jack are two separate events. The probability of getting a heart is 13/52, and the probability of getting a jack is 4/52. Since these events are independent, we can multiply their probabilities: (13/52) * (4/52) = 52/2704 or approximately 0.019.

10. P(heart or a jack): Similar to question 8, to find the probability of the union of two events ("or"), we add their individual probabilities. The probability of getting a heart is 13/52, and the probability of getting a jack is 4/52. Since these events are mutually exclusive (a card cannot be both a heart and a jack), we add the probabilities: (13/52) + (4/52) = 17/52 or 0.327.

I hope this clarifies how to find the probabilities in this scenario. Let me know if you have any further questions!