Introduction to 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:

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!

And Vs. Or Probability Practice:

Example 1

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


Example 2

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



Example 3

If you draw a card from a deck of cards, what is the probability of it being a face card and an ace?



Use this bag of marbles to answer questions 1-3.

bag of marbles


Exercise 1

What is the probability of blindly reaching in the bag and pulling out a green or blue marble?


Exercise 2

What is the probability of pulling out an orange marble or not a blue one?


Exercise 3

What is the probability of pulling out a not-blue and not-orange marble?


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