And vs. Or Probability at a Glance

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:

AND Probability

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.

P= 6/52 = 3/26 ≈ 11.5%

OR Probability

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.

32/52 = 8/13≈61.5%

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!

Examples
Exercises

Example 1

If you roll a die, what is the probability of it being odd and less than 5?

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Example 2

If you roll two dice and find their sum, what is the probability of the sum being even or greater than 8?

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Example 3

If you draw a card from a deck of cards, what is the probability of it being: 1. a face card and an ace? 2. a 3 and a black card? 3. a face card or a black card?
a. The answer to the first question is a bit tricky: there are no cards that satisfy both conditions, so the probability is 0.
b. The second question is more likely. There are two black suits: spades and clubs. In each of these suits, there is only one 3 card. So there are two black cards with a 3 on them in the entire deck. The probability of drawing a black three is or = 3.8%.
Possible, but not very likely.
c. The third question is a lot more likely. How many face cards are there? There are 3 face cards in each suit (jack, queen, king) so that makes 12 face cards in all.
How many black cards are there? Half of the cards are black so there are 26 black cards. But wait, we already counted 6 of the black cards when we added up the face cards, so that leaves 20 black cards.
There are 32 cards that are a red face card, or a black face card, or a black card. That makes the probability , or 61.5%.