Earlier today I set you the following two puzzles: 1. Deck dilemma Your friend chooses at random a card from a standard deck of 52 cards, and keeps this card concealed. You have to guess which of the 52 cards it is. Before your guess, you can ask your friend one of the following three questions:
Your friend will answer truthfully. What question would you ask that gives you the best chance of guessing the correct card? Solution It doesn’t matter. In all three cases, your chance of guessing the correct card is 1 in 26. It’s a wonderful little puzzle because the result seems so counter-intuitive. Any question about the type of card gives you exactly the same help, which is to double your chances of getting the correct card. Case 1. Once your friend replies, you will know if the card is red or black. There are 26 red, and 26 black cards, so you have a 1 in 26 chance of guessing the correct one. Case 2. There is a 12/52 chance the card is a face card, and a 40/52 chance it isn’t. If your friend replies that it is a face card, you have a 1/12 chance of guessing the correct card, and if your friend replies it isn’t, you have a 1/40 chance. Thus the probability of guessing the card when it is a face card is (12/52) x (1/12) = 1/52, and the probability of guessing the card when it isn’t is (40/52) x (1/40) = 1/52. The overall probability of guessing the card is the sum of these two probabilities, which is 1/52 + 1/52 = 1/26 Case 3. The same argument applies. If the card is the ace of spades you will be told this fact by your friend, and this outcome has a 1/52 chance of happening. If the card isn’t the ace of spades, which has a 51/52 chance of happening, you must then choose 1 card from the remaining 51. This outcome gives you a probability of (51/52) x (1/51) = 1/52. Again, the sum of both possible outcomes is 1/52 + 1/52 = 1/26. 2. Heart is in pieces The image below is a spade. Can you cut it into three pieces such that it is possible to reassemble the pieces and make a heart?  To be clear, what you are being asked to do is this: imagine the spade is made of card. Make two cuts to the card, thus cutting it into three pieces, and then reassemble the pieces without overlapping so that the pieces together make the shape of a heart, that is, the symbol of the suit of hearts. The cuts may, or may not, be straight lines. Solution Cut as below, and then turn it around. I hope you enjoyed today’s puzzles. I’ll be back in two weeks. I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me. The first puzzle is adapted from Basic Probability, What Every Math Student Should Know by the eminent Dutch mathematician Henk Tijms. If you are interested in probability, and want an accessible, historical take, you might enjoy the following book by Henk’s son Steven Tijms: Chance, Logic and Intuition: An Introduction to the Counter-Intuitive Logic of Chance. The origin of the second puzzle is either Sam Loyd or Henry Dudeney, who both published the puzzle around 100 years ago. I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.
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Return to Statistics Topics Summary of Basic ProbabilityThe classical or theoretical definition of probability assumes that there are a finite number of outcomes in a situation and all the outcomes are equally likely. Classical Definition of Probability
Though you probably have not seen this definition before, you probably have an inherent grasp of the concept. In other words, you could guess the probabilities without knowing the definition. Cards and Dice The examples that follow require some knowledge of cards and dice. Here are the basic facts needed compute probabilities concerning cards and dice. A standard deck of cards has four suites: hearts, clubs, spades, diamonds. Each suite has thirteen cards: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen and king. Thus the entire deck has 52 cards total. When you are asked about the probability of choosing a certain card from a deck of cards, you assume that the cards have been well-shuffled, and that each card in the deck is visible, though face down, so you do not know what the suite or value of the card is. A pair of dice consists of two cubes with dots on each side. One of the cubes is called a die, and each die has six sides.Each side of a die has a number of dots (1, 2, 3, 4, 5 or 6), and each number of dots appears only once. Example 1 The probability of choosing a heart from a deck of cards is given by
Example 2 The probability of choosing a three from a deck of cards is
Example 3 The probability of a two coming up after rolling a die (singular for dice) is
The classical definition works well in determining probabilities for games of chance like poker or roulette, because the stated assumptions readily apply in these cases. Unfortunately, if you wanted to find the probability of something like rain tomorrow or of a licensed driver in Louisiana being involved in an auto accident this year, the classical definition does not apply. Fortunately, there is another definition of probability to apply in these cases. Empirical Definition of Probability The probability of event A is the number approached by
as the total number of recorded outcomes becomes "very large." The idea that the fraction in the previous definition will approach a certain number as the total number of recorded outcomes becomes very large is called the Law of Large Numbers. Because of this law, when the Classical Definition applies to an event A, the probabilities found by either definition should be the same. In other words, if you keep rolling a die, the ratio of the total number of twos to the total number of rolls should approach one-sixth. Similarly, if you draw a card, record its number, return the card, shuffle the deck, and repeat the process; as the number of repetitions increases, the total number of threes over the total number of repetitions should approach 1/13 ≈ 0.0769. In working with the empirical definition, most of the time you have to settle for an estimate of the probability involved. This estimate is thus called an empirical estimate. Example 4 To estimate the probability of a licensed driver in Louisiana being involved in an auto accident this year, you could use the ratio
To do better than that, you could use the number of accidents for the last five years and the total number of Louisiana drivers in the last five years. Or to do even better, use the numbers for the last ten years or, better yet, the last twenty years. Example 5 Estimating the probability of rain tomorrow would be a little more difficult. You could note today's temperature, barometric pressure, prevailing wind direction, and whether or not there are rain clouds that could be blown into your area by tomorrow. Then you could find all days on record in the past with similar temperatures, pressures, and wind directions, and clouds in the right location. Your rainfall estimate would then be the ratio
To make your estimate better, you might want to add in humidity, wind speed, or season of the year. Or maybe if there seemed to be no relation between humidity levels and rainfall, you might want add in the days that did not meet your humidity level requirements and thus increase the total number of days. Example 6 If you want to estimate the probability that a dam will burst, or a bridge will collapse, or a skyscraper will topple, there is usually not much past data available. The next best thing is to do a computer simulation. Simulation results can be compiled a lot faster with a lot less money and less loss of life than actual events. The estimated probability of say a bridge collapsing would be given by the following fraction
The more true to life the simulation is, the better the estimate will be. Basic Probability Rules For either definition, the probability of an event A is always a number between zero and one, inclusive; i.e.
Sometimes probability values are written using percentages, in which case the rule just given is written as follows
If the event A is not possible, then P(A) = 0 or P(A) = 0%. If event A is certain to occur, then P(A) = 1 or P(A) = 100%. The sum of the probabilities for each possible outcome of an experiment is 1 or 100%. This is written mathematically as follows using the capital Greek letter sigma (S) to denote summation.
Probability Scale* The best way to find out what the probability of an event means is to compute the probability of a number of events you are familiar with and consider how the probabilities you compute correspond to how frequently the events occur. Until you have computed a large number of probabilities and developed your own sense of what probabilities mean, you can use the following probability scale as a rough starting point. When you gain more experience with probabilities, you may want to change some terminology or move the boundaries of the different regions.
*This is a revised and expanded version of the probability scale presented in Mario Triola, Elementary Statistics Using the Graphing Calculator: For the TI-83/84 Plus, Pearson Education, Inc. 2005, page 135. Return to Statistics Topics |
What is the probability that you will select a ace king queen or jack from a deck of 52 cards?
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