h) Are the events spade and three independent events? Why or why not?

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Let

- $$ S $$ be the event that a card is a spade.
- $$ T $$ be the event that a card is a three.

In a standard deck of 52 cards:

1. The probability of drawing a spade is 
   $$
   P(S) = \frac{13}{52} = \frac{1}{4}.
   $$

2. The probability of drawing a three is 
   $$
   P(T) = \frac{4}{52} = \frac{1}{13}.
   $$

3. The probability of drawing a card that is both a spade and a three (only the three of spades qualifies) is 
   $$
   P(S \cap T) = \frac{1}{52}.
   $$

For independence, the condition is 
$$
P(S \cap T) = P(S) \cdot P(T).
$$

Calculating the product of the individual probabilities, we have:
$$
P(S) \cdot P(T) = \frac{1}{4} \cdot \frac{1}{13} = \frac{1}{52}.
$$

Since 
$$
P(S \cap T) = \frac{1}{52} = P(S) \cdot P(T),
$$
the events $$ S $$ and $$ T $$ are independent.

Thus, the events spade and three are independent because the probability of both occurring together equals the product of their individual probabilities.
The events "spade" and "three" are independent because the probability of both a spade and a three occurring together is equal to the product of their individual probabilities.

h) Are the events spade and three independent events? Why or why not?

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