Probability Theory

Probability theory is the branch of mathematics concerned with the likelihood of events occurring. It involves calculating probabilities of simple and compound events, understanding concepts like sample space, conditional probability, expected value, and using combinatorial methods to count outcomes.

  • Question

    Hannah is waiting for a bus. Which of the following three events is most likely to occur:

    1. Hannah waits for the bus for at least one minute,
    2. Hannah waits for the bus for at least two minutes,
    3. Hannah waits for the bus for at least five minutes.

  • Question

    Every evening, Yuval finishes work at a random time and arrives at a bus stop. At this station, two buses stop: number `7`, which goes to Yuval's house, and number `13`, which goes to the house of his friend Shlomi. Yuval gets on the first bus that arrives and, depending on that, goes to Shlomi's or home.

    After a while, Yuval notices that after work, he goes to Shlomi's about twice as often as he goes home. He deduced from this that bus number `13` arrives twice as frequently as bus number `7`.

    Is Yuval necessarily correct?

  • The Confrontation

    Ehud and Benjamin are participating in a public debate. Each one, in turn, presents a question to their opponent. Ehud is chosen to be the first to present a question. A "trick question" is a question to which the opponent has no answer. A contestant who manages to ask a trick question immediately wins the debate. The probability of each of the two contestants finding a trick question (in their turn) is exactly 1/2. It is also known that there is no dependence between the questions. What is the probability of Ehud winning the debate?

  • The Confrontation

    Ehud and Benjamin are participating in a public confrontation. Each presents, in turn, a question to their opponent. Ehud is chosen to be the first to present a question. A "tricky question" is a question for which the opponent has no answer. A contestant who manages to ask a tricky question immediately wins the confrontation. The probability of each of the two contestants finding (in turn) a tricky question is exactly 1/2. It is also known that there is no dependence between the questions. What is the probability of Ehud winning the confrontation?

  • The Confrontation

    Ehud and Benjamin are participating in a public debate. Each one presents a question to his opponent in turn. Ehud is chosen to be the first to present a question. A "tricky question" is a question that the opponent has no answer to. A contestant who manages to ask a tricky question immediately wins the debate. The probability of each of the two contestants finding (in turn) a tricky question is exactly 1/2. Also, it is known that there is no dependence between the questions. What is the probability of Ehud winning the debate?

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  • The Winning Exercise

    Two fair dice (1-6) are rolled, and two calculations are made:

    1. The sum of all 10 visible faces (5 on each die)

    2. The sum of the two top faces of the dice, multiplied by 5

    What is the probability that the result of the first calculation is greater than that of the second calculation?

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  • TWO QUESTIONS IN PROBABILITIES

    There is perhaps no class of puzzle over which people so frequently blunder as that which involves what is called the theory of probabilities. I will give two simple examples of the sort of puzzle I mean. They are really quite easy, and yet many persons are tripped up by them. A friend recently produced five pennies and said to me: "In throwing these five pennies at the same time, what are the chances that at least four of the coins will turn up either all heads or all tails?" His own solution was quite wrong, but the correct answer ought not to be hard to discover. Another person got a wrong answer to the following little puzzle which I heard him propound: "A man placed three sovereigns and one shilling in a bag. How much should be paid for permission to draw one coin from it?" It is, of course, understood that you are as likely to draw any one of the four coins as another. Sources:

  • THE MONTENEGRIN DICE GAME

    It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation. The two players first select two different pairs of odd numbers (always higher than `3`) and then alternately toss three dice. Whichever first throws the dice so that they add up to one of his selected numbers wins. If they are both successful in two successive throws it is a draw and they try again. For example, one player may select `7` and `15` and the other `5` and `13`. Then if the first player throws so that the three dice add up `7` or `15` he wins, unless the second man gets either `5` or `13` on his throw.

    The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance.

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