And vs. Or Probability
In probability, there's a very important distinction between the words and and or.
- And means that the outcome has to satisfy both conditions at the same time.
- Or means that the outcome has to satisfy one condition, or the other condition, or both at the same time.
Let's look at one probability in these two ways:
What is the probability of drawing a card from a deck and it being red and a face card?
For this probability, we need to look at which cards are both red and face cards. There are 6 of these: Jack of Hearts, Queen of Hearts, King of Hearts, Jack of Diamonds, Queen of Diamonds, and King of Diamonds.
What is the probability of drawing a card from a deck and it being red or a face card?
This time the card can be red, or a face card, or both at the same time. There are 26 red cards (6 of which are also face cards). In addition, there are 6 more face cards that are not red: Jack of Clubs, Queen of Clubs, King of Clubs, Jack of Spades, Queen of Spades, and King of Spades. That is a total of 26 + 6 = 32 cards.
Be careful not to just add up the number of face cards (12) with the number of red cards (26). That would give a total of 38 cards, but it would count the red face cards twice.
Notice how much these two probabilities differ. One little word changes the whole problem!