Question

find the probability of drawing an ace or a spade or both from a deck of cards.

Answer 1

Hint: In a deck of cards there are 13 spades and 4 aces. Then we have to find the probability of drawing spades from deck of cards and probability of drawing aces from deck of cards, now there is a probability of getting aces of spades so remove that probability from addition of those two probabilities. Probability of drawing an ace or a spade or both from a deck of card is denoted by \[P(A\cup B)\]

Complete step-by-step answer:
We know that there are 52 cards in total.
Hence, the total number of possible outcomes is 52
 Probability of drawing an ace or a spade or both from a deck of card is
The total number of spades in a deck of cards is 13
The probability of drawing spades in a deck of cards is \[P(A)=\dfrac{13}{52}\]
The total number of aces cards in a deck of cards is 4
The probability of drawing aces in a deck of cards is \[P(B)=\dfrac{4}{52}\]
The total number of aces of spade is 1
\[P(A\cap B)=\dfrac{1}{52}\]
Probability of drawing an ace or a spade or both from a deck of card is
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
= \[\dfrac{13}{52}+\dfrac{4}{52}-\dfrac{1}{52}=\dfrac{16}{52}=\dfrac{4}{13}\]
Probability of drawing an ace or a spade or both from a deck of cards is \[\dfrac{4}{13}\].

Note: From the venn diagram we can obtain the relation that is \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]. Here \[P(A\cup B)\]represents probability of happening of events A and B or probability of happening both.By using the formula we are able to solve the problem. We know that the probability is the ratio of total number of desired outcomes to the total number of all possible outcomes.