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Browse High School Probability
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Coin tossing.
    Odds vs. probability.
    Poker probabilities.
    Rolling dice.

Expected Value [12/09/1997]
What is expected value?

Expected Value [11/07/2001]
I am playing a game where I toss a fair coin. I get a bead if I flip a head and I lose a bead if I flip a tail... to get a total of 5 beads, what is the expected number of coin tosses I must make?

Expected Value of Area of Triangle [05/10/2000]
What is the expected value of the area of triangle APB if AB is the diameter of a circle with radius r, AP is a chord, and X is the angle between AB and AP, uniformly distributed over the interval 0 to pi/2?

Explaining Bayes' Theorem [03/22/1999]
I need an easy-to-understand explanation of Bayes' Theorem to begin my math project for the math fair.

Fair Chance: Three Things, One Coin to Flip [03/12/2003]
How can I choose with equity (p = 1/3) among 3 things (door, person, etc.) having only a (perfect) coin and not infinite time? How can I generate an event with probability p = 1/3 with one perfect coin in a finite number of tosses?

Fair Value of a Coin Toss Game [01/21/2001]
Suppose you play a game where you toss a coin until tails turn up...

Figuring the Odds of 3 Cards [08/07/1997]
In the card game Super Seven, how do you figure the odds against you of receiving three sevens?

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 a Steady-State Probability Matrix [05/12/2000]
Find the steady-state matrix for the likelihood of a guard's being at each of the corners of a rectangular lot, if he is instructed to wait 10 minutes at each corner, and then either stay where he is or move to one of the adjacent corners randomly.

Finding My Car Key on the Second Try [05/29/2000]
There are seven car keys in a box, including mine. I'm going to randomly remove one key at a time, and then try to start my car...

Find the Density Function [01/24/2002]
Suppose X1, X2, X3 are independent and uniform on(0,1). Find the density function of S = X1 + X2 + X3.

Finishing a Best-of-Seven Series [06/19/2009]
A best-of-seven series is currently 2 games to 1 in favor of team A. If the two teams are equally likely to win a given game, what is the probability that A wins the series by reaching 4 games before B does?

The First Ace [06/12/1997]
An ordinary bridge deck of 52 cards is thoroughly shuffled. The cards are then dealt face up, one at a time, until an ace appears. What is the probability that the first ace appears at the kth card or sooner?

Five-Card Straight [12/10/2002]
From a standard 52-card deck you pull out a 5-6-7-8 (suit is irrelevent). You shuffle the remaining 48 cards and deal 3 more. What percentage of the time will you complete the 5-card straight?

Four of a Kind in a 13-Card Hand [03/24/2001]
What is the probability of receiving four-of-a-kind when dealt 13 cards from a regular 52-card deck?

Four of a Kind in Half a Deck of Cards [11/26/1999]
If a full deck of cards is cut in half, what are the chances that at least one type of a card (i.e. Kings, Aces) will have all four suits present?

Full House in Draw Poker [7/28/1996]
A player is dealt 5 cards - a pair of aces and three other unpaired cards. What is the probability of winding up with a full house...

The Function g(x) = gcd(x, 10) [01/14/1999]
Graph the function g(x) = gcd(x,10), where x is a positive integer. What is the range? Assign a probability that g(x) = r for each r in the range.

Functions of Random Numbers [08/26/1998]
If you add or multiply a random number by a non-random number, do you still get a random number?

Game Theory and Payoff Matrices [11/14/1998]
Can you give me a good introduction to game theory? What is a payoff matrix?

Gender Probabilities for Twins [08/27/2006]
I'm pregnant with twins. What are the probabilities of having two girls, two boys, and one of each? Does the fact that they are twins make the probabilities different from two siblings of different ages?

Generating Random Numbers with Specified Probabilities [08/10/2003]
Given an evenly distributed Pseudo Random Number (PRN) generator (with each generated outcome having the same probability), how can I generate outcomes with specified probabilities?

A Geometry and a Logic Problem [3/11/1995]
Problem 1: A cylindrical hole six inches long is drilled straight through the center of a solid sphere. What is the volume remaining in the sphere? Problem 2: The classical stay-switch problem.

Getting More Heads [05/29/2000]
If Bob flips 850 fair coins and Alice flips 851 fair coins, what is the probability that Alice will get more heads than Bob?

Getting One of Each Number (1-6) Rolling N Dice [05/24/2001]
I've been trying to determine various probabilities involving randomly distributing items to a group of six people...

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 coin?

Gnomes in Trouble (Again!) [04/26/2002]
Ten gnomes have been captured by a Sultan, who forces them to solve a puzzle to escape impalement.

Greatest Probability Neither Event Will Happen [02/02/2002]
The probability that event A will happen is 0.54, and the probability that event B will happen is 0.68; what is the greatest probability that neither of the events will happen?

Guessing on True-False Tests [07/08/1999]
If you guess at each answer, are you more likely to answer at least half the questions right on a test with few or with many questions? Does it matter whether there is an even or an odd number of questions?

Horse Racing Odds [02/21/2002]
In horse racing the odds often add up to more than one. Why?

How Do You Know That Events Are Equally Likely? [07/03/2008]
How do you determine whether the outcomes of an event are equally likely? The outcomes of a fair coin toss are considered equally likely so each has a probability of 1/2. But how do you first determine that the probability of getting heads or tails really is equally likely?

Hyper-geometric Distribution [07/29/1999]
Through a mix-up on a production line, 6 defective refrigerators were were shipped with 44 good ones...

Hypergeometric Probability Distribution [08/08/1999]
Three consecutive bulbs are drawn without replacement at random from a box that contains 3 defective light bulbs and 7 good light bulbs. Construct a tree that shows all the possible outcomes.

Inclusive Probabilities [11/06/1998]
Can you explain the formula for inclusive probability: P(A or B) = P(A) + P(B) - P(A and B)?

Independent and Mutually Exclusive Events [01/03/2007]
Can you explain the difference between events that are independent and events that are mutually exclusive and give examples of each?

Independent Events [08/07/1998]
Are the events A and B independent if P(A) = 0.6, P(B) = 0.2, and P(A and B) = 0.12?

Independent Hunters and Lucky Ducks [01/24/2002]
How many ducks would you expect to survive when fired upon by an equal number of hunters who shoot independently of each others' targets, but with deadly aim? Doctor Anthony models this binomial probability problem of independent trials with balls and boxes.

Independent Outcomes OR (AND?) Dependent Ones [08/21/2012]
A student wonders which probability formula applies, and why. Working through the original problem and then a second example, Doctor Peterson provides an overview of probability inclusion, exclusion, independence, and conditions.

Independent, Uniformly Distributed Random Variables [01/10/2002]
Given two independent, uniformly distributed random variables X and Y in {0,1,...,n}, calculate the probability of (X=j I X+Y=k)...

Independent vs. Dependent Events [06/09/2003]
What is the difference between 'independent' and 'dependent' events?

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