How can we use the concept:
We define probability as the number of desired outcomes out of the total number of possible outcomes. For example, let’s say I have a bag of 10 marbles, 3 of which are red. I want to know the probability of randomly selecting a red marble out of the bag. So the number of desired outcomes is 3 - there are 3 red marbles I could possibly pull out of the bag. The total number of possible outcomes is 10 - there are 10 marbles that I could possibly pull out of the bag. So the probability of selecting red is
or .3. I can interpret that probability by noting that it is closer to 0 than 1 - meaning that the probability of picking a red marble is less likely (though not impossible!).
There’s a couple of important notes to make about the above situation. First of all, note that I wanted the probability of randomly picking a red marble out of the bag. Randomness is important in probability. If I’m allowed to look in the bag (assuming I’m not color blind) I can definitely pick a red marble. The probability is the chance of the event randomly occurring. Also please note that, in the marble situation, it is assumed that each outcome is equally likely. When I reach into the bag to pull out a marble, each of the 10 marbles have an equal chance of being drawn; there is nothing that gives one marble an advantage over the other. In order to do a probability calculation, it is important that each of the total outcomes have an equal chance of occurring.
There’s a couple of important notes to make about the above situation. First of all, note that I wanted the probability of randomly picking a red marble out of the bag. Randomness is important in probability. If I’m allowed to look in the bag (assuming I’m not color blind) I can definitely pick a red marble. The probability is the chance of the event randomly occurring. Also please note that, in the marble situation, it is assumed that each outcome is equally likely. When I reach into the bag to pull out a marble, each of the 10 marbles have an equal chance of being drawn; there is nothing that gives one marble an advantage over the other. In order to do a probability calculation, it is important that each of the total outcomes have an equal chance of occurring.
Sample Math Problems
1. Myra rolls a standard dice with sides labeled 1-6. What is the probability of rolling a 5?
Solution: Remember that probability is the number of desired outcomes out of the total number of possible outcomes. In this case, there is 1 desired outcome - rolling the number 5. There are 6 possible outcomes - all the sides of this dice. Therefore the probability is ⅙.
2. What is the probability of drawing a red king out of a standard deck of cards? Report your answer as a lowest terms fraction.
Solution: In a standard deck of cards, there are two red kings - the king of diamonds and the king of hearts. So there are two desired outcomes. There are 52 possible outcomes - there are 52 cards in a standard deck. So the probability is
. This is not in lowest terms though - I can take a 2 out of the top and bottom of this fraction. So the final answer is
3. Breslin has 50 classmates in the 4th grade and he surveyed them about their favorite ice cream flavor. The following table summarizes his results:
What is the probability of Breslin randomly selecting a classmate who likes vanilla?
Solution: The number of desired outcomes is 15 - there are 15 classmates who like vanilla. From the beginning of the problem, we know that this is out of 50 total classmates. So the probability is
. Let’s reduce this to lowest terms, so the final answer is
4. Researchers studied a group of people to see if late night snacking had any effect on dreaming. The results were as follows:
What is the probability that a person had a dream, given that they ate a late night snack?
Solution: The way we calculate probability has not changed - it will still be the number of desired outcomes out of total possible outcomes. However, we need to consider carefully what those are in this situation. For the number of desired outcomes, note that I want the probability of a person having a dream given that they ate a late night snack. In other words, I want the people who had both - a late night snack and a dream. That would be the top left box of the chart. So our number of desired outcomes is 31 - thirty-one people had a dream and a late night snack. Now for the total number of possible outcomes. I want the probability of having a dream given that they ate a late night snack. In other words, I’m not interested in anyone who didn’t eat a late night snack. So I will disregard the entire right column and instead add up the two numbers in the left column. 31 + 19 = 50. So 50 people ate a late night snack - this is the pool of people that I’m picking out of for this probability. So the probability is
