Probability (Multiple events: And vs. Or)
Often, when we are trying to find the probability of something happening, we want to find the probability of several events happening. To find the probability of multiple events, we have to combine the probability that each event will happen. Exactly how we combine the probabilities depends on exactly what we're trying to figure out!
Before we get into the rules of probability, let's think logically about how different types of questions might affect the way we calculate probabilities.
Let's go back to the example of rolling dice. If we want to know the probability of getting a 3 on one die, we take the number of outcomes we want and put it over the number of total possible outcomes for a probability of $\dfrac{1}{6}$.
If we roll two dice, then we can ask two different types of questions, AND questions and OR questions.
We could ask: What is the probability of getting a 3 on one die AND a 3 on the other die?
Or, we could ask: What is the probability of getting a 3 on one die OR a 3 on the other die?
Let's say you're playing a game where you get $100 if you get a 3. With just one die, you have a $\dfrac{1}{6}$ chance of winning $100 every time you roll the die.
Do you have better or worse chances of winning the $100 if you have to get a 3 on the first die AND a 3 on the second die? Your chances are worse. If you have to meet more than one condition, your probability gets smaller.
Now, what if you win $100 if you get a 3 on the first die OR a 3 on the second die? Now, your odds went up! You just got two chances to get a 3 instead of only one.
So, if your odds of getting a 3 on one die in one roll is $\dfrac{1}{6}$, your chances of getting a 3 on the first die AND on the second die are LESS THAN $\dfrac{1}{6}$, and your chances of getting a 3 on the first die OR the second die are GREATER THAN $\dfrac{1}{6}$.
As we begin to learn how to calculate AND and OR probabilities, always remember to think about whether your chances should get lower or higher, depending on the additional conditions. Thinking that logical step through will help you avoid errors.
Practice Problems:
Probability (Multiple Events: And vs. Or)
State whether the following questions are AND or OR probabilities and if adding the additional conditions will raise or lower the probability. (Note, for practice you can calculate most of these probabilities, but some of the AND probabilities are a little complicated because there is some overlap between the conditions, which is unknown, and which you would typically need to subtract out.)
A local animal shelter knows that the probability that someone who walks into the shelter will adopt a dog is 10% and the probability that someone who walks into the shelter will adopt a cat is 12%. The probability that someone who walks into the shelter will adopt a rabbit, bird, or other animal is 2%.
1. What is the probabilty that someone adopts a dog and a hamster?
2. What is the probability that someone adopts a dog or a cat?
3. What is the probability that someone adopts a dog, cat, and a bird?
4. What is the probability that someone adopts any animal at all?
A fast food restaurant, Texan Burger, is famous for its signature BBQ burger and 60% of customers order a BBQ burger whenever they come in. They also make great fries and 80% of customers order fries when they eat. Nearly 90% of customers order a soft drink whenever they eat at Texan Burger. Just under 15% of customers get a kids meal and 20% of customers order some kind of chicken dish.
5. What is the probability of a customer ordering a chicken dish or a kids meal?
6. What is the probability of a customer ordering a drink and not a BBQ burger?
7. What is the probability of a customer ordering a BBQ burger and either a drink or fries?
8. What is the probability of a customer ordering neither fries nor a drink?
Mariana is learning to play poker. She knows that there are 52 cards in a deck, 4 cards of each type, 13 cards of each suit (e.g., each deck contains 4 kings, 4 queens, and 4 sevens as well as 13 hearts, 13 spades, etc.). She knows that pairs (e.g., 2 aces, 2 eights) are good, but that flushes (all five cards the same suit -- e.g., all five cards are hearts or diamonds) are better. She knows that straights (all 5 cards going in order: 2, 3, 4, 5, 6) are even better. She wants to win!
9. Her first four cards are a 5, a 7, a jack, and an ace. What is the probability that her fifth card will be a pair with one of her first four cards?
10. She really wants either a jack or a heart. What is the probability that she'll get either one?
11. What is the probability that she'll get a jack of hearts?
12. What is the probablity that she'll get a card that is neither a jack or a heart?
Answer Key: