A fair coin is tossed, and a card is drawn from a standard deck (52 cards). What is the probability that you tossed a head or drew an ace?
Let 'A' be the event of getting head from the coin toss and 'B' be the event of getting ace from the deck.
My approach is to use the sum rule, which is
P(A or B) = P(A) + P(B) – P(A ∩ B)
My question is
Since tossing a coin and drawing a card are two independent events, can we take take P(A ∩ B) as zero?
Best Answer
The complement event to tossing a head $\color{red}{\text{or}}$ drawing an Ace is :
Tossing a tail and not drawing an Ace.
Probability of complement event occuring is
$$\frac{1}{2} \times \frac{12}{13} = \frac{6}{13}.$$
Therefore, probability of original event is $1$ minus the probability of the complement event, which is
$$1 - \frac{6}{13} = \frac{7}{13}.$$
In general, the assumption that $A$ and $B$ are two independent events does not imply that $p(A \cap B) = 0.$ Simple counter example: Toss a coin twice. Each coin toss is an independent event. If you let $A$ denote the event that first coin toss is Heads and $B$ denote the event that second coin toss is Heads, you do not have that $p(A\cap B) = 0.$ That is, the probability of getting two consecutive Heads on two coin tosses is not zero.