# Consider of drawing one card from a deck of $52$. Prove that the events of a spade being drawn and an ace being drawn are independent events.

Consider of drawing one card from a deck of $$52$$. Prove that the events of a spade being drawn and an ace being drawn are independent events.

Let $$A$$ be the event that a spade is drawn and let $$B$$ be the event that an ace is drawn.

Then, $$\text P(A) = 4/52 = 1/13$$ and $$\text P(B) = 4/52 = 1/13$$.

How can I calculate $$\text P(A\cap B)$$? And how can I prove that these events are independent since the question specifically asked to prove that they are independent?

• What card did you get in $A \cap B$? What is the probability that you drew that card? What is the definition of independence? – Ethan Bolker Jan 2 at 16:32
• $P(A\cap B)$ is the probability that the card drawn is both an ace and a spade. – Lord Shark the Unknown Jan 2 at 16:32
• Independence is determined by $\text P(A\cap B)=\text P(A)\times\text P(B)$ – TheSimpliFire Jan 2 at 16:33
• In a deck of 52 cards there should be more than four spades be present ... – Michael Hoppe Jan 2 at 16:33
• Is there really a $1/13$ probability of drawing a spade? – Lord Shark the Unknown Jan 2 at 16:33

You didn't compute the probability of event $$A$$ correctly. There are $$13$$ spades in a standard deck. So $$P(A)=13/52=1/4$$. Note that $$A\cap B$$ corresponds to drawing the ace of spades and hence $$\frac{1}{52}=P(A\cap B)=\frac{1}{4}\times\frac{1}{13}=P(A)P(B)$$