f) What is the probability of picking a spade given that the card is a three? Round answer to 3 decimal places. $$ \square $$

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1. We are asked to find the probability of picking a spade given that the card is a three. Using conditional probability, we have
   $$
   P(\text{spade} \mid \text{three}) = \frac{P(\text{spade and three})}{P(\text{three})}.
   $$

2. There are 52 cards in a deck. There are 4 threes in total (one in each suit), so
   $$
   P(\text{three}) = \frac{4}{52}.
   $$

3. Among the 4 threes, only the three of spades is a spade, so
   $$
   P(\text{spade and three}) = \frac{1}{52}.
   $$

4. Substitute these probabilities into the conditional probability formula:
   $$
   P(\text{spade} \mid \text{three}) = \frac{\frac{1}{52}}{\frac{4}{52}} = \frac{1}{4}.
   $$

5. Calculating the value gives:
   $$
   \frac{1}{4} = 0.25.
   $$

6. Rounding to 3 decimal places, the probability is:
   $$
   0.250.
   $$

$$\boxed{0.250}$$
The probability of picking a spade given that the card is a three is 0.250.

f) What is the probability of picking a spade given that the card is a three? Round answer to 3 decimal places. \( \square \)

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