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Problems With Probability
Learning how to solve probability problems can be tricky for many people. Even though the calculations themselves are very simple (basic addition and multiplication), the sequence of math equations is often long and confusing. Let's break down the problem with a little review.
Probability is the chance that an event will occur. Events may be any occurence such as:
- Going to the doctor,
- Standing on a yellow brick or
- Seeing a monkey.
As you can guess, the chance of these events happening changes based on other things that are going on at the time. These "other things" are called variables. For example, if you chip your tooth, it is more likely (more probable) that you will be going to the doctor. If you are in the kitchen, it is less likely (less probable) that you will see a monkey. (Can you think of a variable that would make it more likely that you will be standing on a yellow brick?)
How likely an event is can be described as a number. This number may be written as a ratio (a part to a whole) or a percentage (a part of 100 parts).
Let's illustrate this concept with a familiar model: the coin toss.
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The Coin Toss
When you flip a coin, there are two events can occur:
- The coin could land on heads or
- the coin could land on tails.
Since there is no way to land on both sides at the same time, only 1 of these events can occur.
Problem 1: So what is the probability of landing on heads?
Heads is 1 out of 2 options.
So the probability of getting a heads is 1 out of 2--the same goes for tails.
1 out of 2 may be written as the ratio (1:2) or a percentage (50%).
Slow down! Let's look at the steps we just completed.
- Count the number of possible events. (2)
- Decide which event you are examining for probability. (Heads)
- Count the number of chances that heads can occur out of the possible events (1)
- Write the number of chances heads could occur over the number of possible events in a ratio. (1:2)
But what does that probability mean?
Having a 1:2 probability for heads means that you will get a heads half of the time. This is why coins are used to make decisions, like who goes first in a football game--both teams have a 50% chance of going first and that is fair.
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Want to Test the Probability?
A probability tells you how likely something is to occur. This doesn't mean that an event is guaranteed to happen, just if it is more or less likely to occur.
As we know from the coin model, we have a 50% chance of getting a heads on every toss. This probability doesn't change no matter how many times we toss the coin. And we can test the probability easily--just toss a coin.
Coin Toss Experiment
Materials Needed: A paper, a pencil and a quarter
- Toss the quarter 100 times and tally the number of heads and tails.
- Count the number of tallies for each event.
- The number of each event occurring will be very close to 50. (For example, 45 heads and 55 tails)
- If the numbers of heads and tails are NOT close to 50%, toss the quarter 50 more times. The more you toss the quarter, the more likely you are to have an even distribution.
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A standard die, the kind you would use for a board game, offers 6 potential events. (Can you guess what they are?)
That's right! Landing on each side of the die is an event. You could roll a 1, 2, 3, 4, 5 or 6.
It can be confusing doing probability problems with die because the sides are numbered. Make it easier to keep the numbers straight by writing out the number when referring to a side of the die. For example write "three" instead of "3."
Problem 2: What is the probability of rolling a 4? Use the four steps outlined above to write figure out this probability and write it as a ratio.
- Count the number of possible events. There are 6 sides to the dice. So there are 6 possible events.
- Decide which event you are examining for probability. The problem let's us know we are trying to roll a four.
- Count the number of chances that heads can occur out of the possible events. There is only one side of the die that has 4 dots, so there is only 1 chance to roll a four out of 6 total chances.
- Write the number of chances heads could occur over the number of possible events in a ratio. (1:6)
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1. Even though there are only 3 different colors, dividing the circle into even sections makes handling the probability easier. This way there are four equally possible outcomes. (white, white, black or orange)
2. We are looking at the probability of landing on black.
3. Black is 1 out of four options.
4. So the probability of landing on black is 1:4 or 25%.
Problem 4: What is the probability of landing on a white area?
- There are four possible options.
- We are figuring out the probability of landing on white.
- Two out of four options are white
- So the probability of landing on white is 1:2! (Did you think it was 2:4? Always remember to simplify your ratios!)
Some more methods are explored the next page...
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There are many different types of probability that describe the circumstances, the variables, that impact a certain event. A joint probability is the chance of two events happening back to back.
Follow these steps to solve a joint probability.
- Write down the probability of the first event. (Just follow the four-step process we used earlier.)
- Write down the probability of the second event.
- Multiply the two ratios.
Problem 5: What is the probability of tossing two heads in a row?
Since we already did the math, we know that the probability of tossing a heads is 1/2.
We also know that this doesn't change. No matter how many times we flip the coin, there will always be two options, one of which is heads. So the chance of tossing a heads is still 1/2.
(1/2) * (1/2) = ?
When multiplying fractions, multiply the numerators (top numbers) and then the denominators (the bottom numbers). Don't forget to simplify the product.
1 * 1 = 1
2 * 2 = 4
So the product is 1/4. There is a 1:4 or 25% chance of getting two heads in a row.
Do You Know What That Means?
The probability of flipping heads once is greater than the probability of flipping heads twice! When we try to get two events to happen back to back, in a sequence, we lower the probability.
Can you guess what happens when we try to get three events to happen?
The probability is even lower...
Try to figure out the probability of getting three heads in a row.
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Practice using the steps to solve the following probability problems. Refer back to the previous examples for help. If you get stuck, take a deep breath and start over with step 1.
Keep in mind that these are word problems. List the given and needed information before attempting to solve the problem. That way you won't miss important details.
Problem 6: There are two tokens in a bag, one is white and the other is blue. What is the probability that you will take out the blue token? (Hint: Coin Toss)
Problem 7: Your mother makes turkey on weekdays and beef on the weekends. You don't know what day it is. What is the probability that you will have turkey for dinner? (Hint: Circle)
Problem 8: There are two puppies at Shelly's house. Shelly hopes that the puppies are girls. What are the chances that one puppy is a girl? What are the chances that both puppies are girls?
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Practice Problem Answers
Answer 6: There are two different events, and you can only have white or blue--just like the coin can only show heads or tails. There is a 1:2 chance that you will draw the blue token.
Answer 7: This is a tricky problem because of the wording. You know that your mother makes turkey on 5 days, and beef on 2. Even though there are only two options, it is easier to solve the probability problem by keeping the week divided into equal pieces--7 days. Turkey is dinner 5 out of 7 days. So the probability of having turkey is 5:7.
Answer 8: This was a two-part question. First you want to know what the chances are of one puppy being a girl. Even though there are two puppies, we are only thinking about one right now. That one puppy can either be a boy or a girl. That makes two possible events, with one outcome; so there is a 1:2 chance that one puppy is a girl. The second puppy has the same chance. So using the steps for solving a joint probability there is 1:4 change that both puppies are girls.
Another way to look at this is to write out the possible gender combinations.
Four options. "Girl-girl" is 1 out of 4 possible events.
If you know that one puppy is a girl. What is the probability that they are both girls? (The answer is not 1:2)
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Need More Help on How to Solve Probability Problems?
- Ask a teacher. Your teachers are there to help you out. If you don't understand something in class, raise your hand and ask. If it's homework that has you stumped, do your best to get it done. Then talk with your teachers the next time you see them. They'll appreciate the effort, and be more than happy to lend a hand in-class or after-school.
- Work with a friend. Two heads really are better than one. Work through your homework problems with a classmate (unless your teacher asked you not to).
- Call the Homework Hotline. They provide free homework help over the phone. A tutor will listen to your problem and help you find AND UNDERSTAND the right answer. DIAL-A-TEACHER HOMEWORK HELP is open Monday-Thursday, 4:00pm to 7:00pm. Just call 585-262-5000 if you live in the Rochester, NY area or 1-888-986-2345 toll free (everywhere else).
- Visit one of the sites below for more examples of probability problems. Sometimes it helps to read two or three different explanations of how to solve a problem. You never know which one is going to "click" and make it clear for you.
- Cut-the-knot.org/hall.shtml - The Monty Hall problem is the focus of this page.
- Homework-hotline.org - This is the website of the Dial-A-Teacher Homework Hotline. This website has additional resources for parents and teachers.
- Math.youngzones.org/joint.html - This site focuses on joint probability problems.
- Cut-the-knot.org/bears.shtml - The inspiration for the puppy problem.
- Probability Lesson Plan: Teach Probability With Examples - This article is a great advanced resource for teaching and learning probability.
Photo Credits: "Circle" by Sylvia Cini
"Die" by Sylvia Cini
"Puppies" by Eduardo Marquetti/Flickr