Intuitive Probability

One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it.
(Laplace, 1825)

The theory of probability supplies a good deal of counterintuitive results. (See, for example, Bear cubs problem, Birthday Coincidence, Lewis Carroll's pillow problem, Monty Hall Dilemma, but there are more.) However, the theory of probability arose from practical applications and is, in essence, a formal encapsulation of the intuitive view on chance.

This page presents a collection of probability problems whose solution is based on intuition and common sense and does not require any theoretical knowledge.

  1. Which situation is more likely after four bridge hands have been dealt: you and your partner hold all the clubs or you and your partner have no clubs? (A game of bridge is played by two pairs of players, with players on a team sitting opposite each other) [Gardner, p. 39].

    Solution

  2. A secretary types four letters to four people and addresses the four envelopes. If she inserts the letters at random, each in a different envelope, what is the probability that exactly three letters will go into the right envelope? [Gardner, p. 39].

    Solution

  3. A pencil with pentagonal cross-section has a maker's logo imprinted on one of its five faces. If the pencil is rolled on the table, what is probability that it stops with the logo facing up? [Winkler, p. 1].

    Solution

  4. Find the probability that if the digits 0, 1, 2, ..., 9 be placed in random order in the blank spaces of 5_383_8_2_936_5_8_203_9_3_76 the resulting number will be divisible by 396.

    Solution

  5. Half the people who contracted a certain disease which is spreading across the country have died and half got better on their own.

    Two medicines have been developed, A and B, but not actually tested. A was administered to three patients, and all survived. B was administered to 8 patients of which seven survived.

    In the unfortunate event of your contracting the disease, which medicine would you choose?

    Solution

References

  1. M. Gardner, The Colossal Book of Short Puzzles and Problems, W. W. Norton & Company, 2006
  2. P. Winkler, Mathematical Mind-Benders, A K Peters, 2007

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Copyright © 1996-2018 Alexander Bogomolny

If there is no magic or cheating, the probabilities of the two teams of getting any specific combination of cards must be thought equal. Now, observe that the event "I and my partner got no clubs" is the same as "The other team got all the clubs". The latter event has the same probability as "Me and my partner got all the clubs," implying that the two events in question are equiprobable, i.e., have equal probabilities.

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Copyright © 1996-2018 Alexander Bogomolny

It is impossible for the pencil to stop on en edge as to make a face - with or without the maker's logo - show on top. The probability of the event in question is zero.

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Copyright © 1996-2018 Alexander Bogomolny

Since there were just four letters and four envelopes, the fact that some three letters were placed into the right envelopes implies that the remaining letter has also been placed into the right envelope. Therefore the probability is 0 that exactly three letters went into the right envelope.

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Copyright © 1996-2018 Alexander Bogomolny

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