e) What is the probability of picking a spade? Round answer to 3 decimal places. $$ \square $$ f) What is the probability of picking a spade given that the card is a three? Round answer to 3 decimal places. $$ \square $$ g) What is the probability of picking a three given that the card is a spade? Round answer to 3 decimal places. $$ \square $$ h) Are the events spade and three independent events? Why or why not? Select an answer

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**e)** Total number of spades in a deck is $$13$$, and total cards is $$52$$. So, the probability is  
$$
P(\text{spade}) = \frac{13}{52} = 0.25 \quad (\text{or } 0.250 \text{ rounded to 3 decimal places}).
$$

**f)** There are $$4$$ cards that are threes. Only one of them is a spade. Thus, the probability of picking a spade given that the card is a three is  
$$
P(\text{spade} \mid \text{three}) = \frac{1}{4} = 0.25 \quad (\text{or } 0.250 \text{ rounded to 3 decimal places}).
$$

**g)** There are $$13$$ spades in the deck and only one of them is a three. Therefore, the probability of picking a three given that the card is a spade is  
$$
P(\text{three} \mid \text{spade}) = \frac{1}{13} \approx 0.0769 \quad (\text{or } 0.077 \text{ rounded to 3 decimal places}).
$$

**h)** Two events $$A$$ and $$B$$ are independent if  
$$
P(A \cap B) = P(A)P(B).
$$
Here,  
$$
P(\text{three}) = \frac{4}{52} = \frac{1}{13}, \quad P(\text{spade}) = \frac{13}{52} = \frac{1}{4},
$$
and  
$$
P(\text{spade} \cap \text{three}) = \frac{1}{52}.
$$
Calculating the product of the two individual probabilities:  
$$
P(\text{three})P(\text{spade}) = \frac{1}{13} \cdot \frac{1}{4} = \frac{1}{52}.
$$
Since  
$$
P(\text{spade} \cap \text{three}) = P(\text{three})P(\text{spade}),
$$
the events spade and three are independent.

Thus, the answer to part h is: Yes, the events are independent because the probability of their intersection equals the product of their individual probabilities.
**e)** The probability of picking a spade is 0.250. **f)** The probability of picking a spade given that the card is a three is 0.250. **g)** The probability of picking a three given that the card is a spade is 0.077. **h)** Yes, the events spade and three are independent because the probability of their intersection equals the product of their individual probabilities.

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