What is the probability that a card drawn from a deck
is either red or an ace ?
Is drawing an ace independent of drawing a red card ?
Since half of the 52 cards are red, and there are 2 aces that
are not red,
the answer is obviously ( 26 + 2 ) / 52 = 7 / 13.
Now let's find out if getting a red card is independent of getting an ace.
Let A be the event that the card selected is an ace.
Let R be the event that the card selected is red.
The probability of A and R is the probability of drawing a red ace,
and this is 2 / 52 = 1 / 26.
i.e. P( A and R ) = 1 / 26.
P( A ) = 4 / 52 = 1 / 13
P( R ) = 1 / 2
Now we can check to see if the product rule yields a true statement.
P( A ) * P( R ) =? P( A and R )
P( A ) * P( R ) =
( 1 / 13 ) * ( 1 / 2 ) = 1 / 26
= P( A and R ).
So the events are indeed independent.
Let's also check the sum rule:
P ( A or R ) =? P( A ) + P( R ) - P( A and R )
| P( A ) + P( R ) - P( A and R ) | = | ( 1 / 13 ) + ( 1 / 2 ) - ( 1 / 26 ) | |
| = | ( 2 / 26 ) + ( 13 / 26 ) - ( 1 / 26 ) | ||
| = | 14 / 26 = 7 / 13 | ||
This agrees with our initial calculation of P( A or R ).