Question

# From a standard 52-card deck, what is the probability that a drawn card is a club or anything besides a face card?

Upper level probability
From a standard 52-card deck, what is the probability that a drawn card is a club or anything besides a face card?

2021-05-04

Clubs account for 1/4 of the cards in the deck. Face cards (Jack, Queen King) account for $$\displaystyle\frac{{3}}{{13}}$$ cards in each suit, leaving $$\displaystyle\frac{{10}}{{13}}$$ per suit.
The probability of the drawn card is as follows: $$P(Clubs) \bigcup P(Facecard)' P(Clubs) =$$ $$\displaystyle\frac{{13}}{{52}}=\frac{{1}}{{4}}$$
$$P(Facecard)' =$$ $$\displaystyle\frac{{40}}{{52}}=\frac{{10}}{{13}}$$
The union of the two possibilities is the possibility of the first, or the the second happening. If the possibilities overlap (a 6 of clubs for example), then that possibility is deducted in the second probability that is added.
$$P(Clubs) \bigcup P(Facecard)'=13/52 + (40/52 - 10/52) = 13/52 + 30/52 = 43/52$$
Your answer is the sum of the clubs suit and the remaining non-face cards in the deck. $$\displaystyle\frac{{43}}{{52}}$$ cards