See also the
Dr. Math FAQ:
probability in the
3D and higher
Browse Middle School Probability
Stars indicate particularly interesting answers or
good places to begin browsing.
See also Middle School Statistics.
Selected answers to common questions:
- Drawing Marbles [02/01/1997]
A jar contains 2 red, 3 blue, and 4 green marbles. Niki draws one marble
from the jar, and then Tom draws a marble. What is the probability that
Niki will draw a green marble and Tom will draw a blue marble?
- Gambler's Fallacy [03/14/2003]
My co-worker prefers to bet the same set of five lottery numbers every time, but I say that the probability is the same if you randomly select any set five numbers for the same period of time. Who is right?
- Math Symbol for C [12/15/1997]
I am puzzled by one symbol of typing math. What does the upper case
letter C mean? As in (2C1) (3C1) / (47C2) = 6/1081.
- Odds Slang for Probability? [02/06/1997]
Is "odds" slang for probability?
- Probability of Two Dice Summing to 5 [09/22/2001]
If a person rolls two dice, what is the probability of getting a five as
- Probability: Permutations and Combinations [12/10/2002]
How is probability related to permutations and combinations?
- About Probability [6/7/1995]
I am wondering how probability works.
- Binomial Experiments [09/05/2003]
How can I explain the binomial distribution to my six grade math class?
- Birthdays Shared across Generations, in Context [08/17/2015]
A grandmother wonders about the unlikelihood of the birthdays of her grandchildren
— one born on her birthday, the other born on her husband's. Confirming the
long odds, Doctor Ali puts the probability into context.
- Calculating Pi - the Nail Drop Experiment [05/19/1999]
Do you have any information about calculating pi by means of the nail
- Cascading Coffee Confusions [09/17/2017]
A teen and her family disagree about the likelihoods of her coffee order going awry.
Drawing out several models, Doctor Ian clarifies how to properly weight two outcomes
from three distinct sequences of events.
- Coin Flipping [01/26/1998]
How can I figure out the chances of flipping a coin five times with the
- Dart Data [02/29/2012]
A mother and daughter struggle to enumerate possible scores from throwing four
darts at three scoring rings. Doctor Peterson suggests another way to tabulate the
tosses: methodically organize the data into groups of four outcomes rather than
treating each dart individually.
- Defining Probability [4/30/1996]
Is a probability number equal to a percent or a decimal?
- Drawing Aces [06/12/1997]
What is the probability that you will pick two aces in a row out of a 52-
- Equally Likely vs. Equally Possible [03/04/2010]
A parent tries to understand her son's homework: "When a baby is born, it is either
right-handed or left-handed. Are these possibilities equally likely? Decide and
- Factorial [4/1/1996]
What is factorial?
- Finding and Interpreting the Odds of Success [08/04/2004]
What does it mean when you see odds expressed as 5 to 1 (or 5:1)? How
do you calculate such odds?
- Finding Probability on the Web [5/15/1996]
I need a good history of probability, theory and basic concepts.
- Four Safety Match Tips, One General Probability Tip [06/14/2015]
An adult struggles to make heads or tails of a probability problem. Modeling it as a
more familiar question of coin tosses, Doctor Ian lights a way forward.
- Geometric Probability [05/23/2001]
If an arrow is equally likely to hit anywhere inside a circular target
that is 3 feet in diameter, what is the probability that it will hit
inside a bull's-eye that is 6 inches in diameter?
- Getting Two Heads in Four Tosses of a Coin [05/17/2000]
What is the probability of getting two heads on four flips of an unbiased
- History of Probability [04/07/1997]
Who first researched probability?
- Independent vs. Dependent Events [06/09/2003]
What is the difference between 'independent' and 'dependent' events?
- Infinite Probability: A Point on a Line [11/03/2004]
Imagine a line extending infinitely in both directions. A line segment
of length 10m has endpoints at point A and point B, both of which are
on the line. What is the probability that a randomly chosen point on
the line falls on line segment AB?
- Lottery vs. Lightning: Shock and Odds [01/09/2011]
A man struck by lightning on four separate occasions wants to know how rare that makes him -- and how such long odds compare with winning the lottery. Doctor Wallace obliges, drawing on data from the National Lightning Safety Institute, and treating the accidents as independent events.
- Math in Card Games [10/06/2000]
How is math involved in creating and playing card games?
- Monkeys Typing Shakespeare: Infinity Theory [08/05/1998]
Would an infinite number of monkeys typing at random eventually produce
the entire works of Shakespeare?
- Most Frequently Rolled Number [10/27/1999]
What sum would be rolled most often if two dice, each numbered -3, -2, -
1, 0, 1 and 2, were rolled 100 times?
- Penny Toss [12/17/1997]
Three people each toss a penny at the same time. What is the probability
that two people get the same side of the penny and the other person gets
the opposite side?
- Pie Graphs and Probability [09/21/2000]
I don't know how to get probability from a pie graph.
- Popsicle Probability [02/27/2003]
There are 9 popsicles: 3 orange, 3 cherry, 3 grape. There are 4
children. What is the probability that all 4 children will get the
flavor of their choice?
- Probability and Odds [01/05/2008]
What is the difference between giving the probability that something
will happen and the odds that it will happen?
- Probability and the 'Ways Method' [10/25/2002]
Suppose we roll one six-sided die. What are the possible outcomes?
What is the probabiliy of rolling a 4? If we have two dice how many
outcomes are there? With two dice what is the probability of rolling a
- Probability of Divisibility [06/18/2002]
What is the probability that a randomly selected three-digit number
is divisible by 5?
- Probability of Even vs. Odd Sums [10/11/2000]
My son believes that addition problems are more likely to have even
answers because there are more ways to get an even sum than an odd sum.
Can you explain to him why that isn't true?
- Probability of Getting your Hat Back [12/9/1995]
There are 5 people with 5 hats. The hats are put into a box. What is the
probability that each person will get his or her hat?
- Probability of Matching Times on a Clock [09/14/2004]
What is the probability of two different times within the same hour
ending in the same last digit, such as 8:13 and 8:43?
- Probability of Picking Coins [01/12/2003]
Max has 5 coins in his pocket that total 47 cents. What is the
probability that he will reach into his pocket and pull out a dime,
and then without replacing it reach in and pull out a quarter?
- Probability Problem [6/4/1995]
Given a box with 12 letters, one of which is a
D, and 2 are E's. I know the probability of getting a D is therefore 1
out of 2, and E is 1 out of 6. How do I determine the
probability of getting a D and then an E, if the D is replaced after