What is the probability that the card drawn from a standard 52 card deck is a spade given that the card is black?
|
Earlier today I set you the following two puzzles: Show 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. Playing cards probability problems based on a well-shuffled deck of 52 cards. Basic concept on drawing a card: In a pack or deck of 52 playing cards, they are divided into 4 suits of 13 cards each i.e. spades ♠ hearts ♥, diamonds ♦, clubs ♣. Cards of Spades and clubs are black cards. Cards of hearts and diamonds are red cards. The card in each suit, are ace, king, queen, jack or knaves, 10, 9, 8, 7, 6, 5, 4, 3 and 2. King, Queen and Jack (or Knaves) are face cards. So, there are 12 face cards in the deck of 52 playing cards. Worked-out problems on Playing cards probability: 1. A card is drawn from a well shuffled pack of 52 cards. Find the probability of: (i) ‘2’ of spades (ii) a jack (iii) a king of red colour (iv) a card of diamond (v) a king or a queen (vi) a non-face card (vii) a black face card (viii) a black card (ix) a non-ace (x) non-face card of black colour (xi) neither a spade nor a jack (xii) neither a heart nor a red king Solution: In a playing card there are 52 cards. Therefore the total number of possible outcomes = 52 (i) ‘2’ of spades: Number of favourable outcomes i.e. ‘2’ of spades is 1 out of 52 cards. Therefore, probability of getting ‘2’ of spade Number of favorable outcomes = 1/52 (ii) a jack Number of favourable outcomes i.e. ‘a jack’ is 4 out of 52 cards. Therefore, probability of getting ‘a jack’ Number of
favorable outcomes = 4/52 (iii) a king of red colour Number of favourable outcomes i.e. ‘a king of red colour’ is 2 out of 52 cards. Therefore, probability of getting ‘a king of red colour’ Number of favorable outcomes = 2/52 (iv) a card of diamond Number of favourable outcomes i.e. ‘a card of diamond’ is 13 out of 52 cards. Therefore, probability of getting ‘a card of diamond’ Number of favorable outcomes
= 13/52 (v) a king or a queen Total number of king is 4 out of 52 cards. Total number of queen is 4 out of 52 cards Number of favourable outcomes i.e. ‘a king or a queen’ is 4 + 4 = 8 out of 52 cards. Therefore, probability of getting ‘a king or a queen’ Number of favorable outcomes = 8/52 (vi) a non-face card Total number of face card out of 52 cards = 3 times 4 = 12 Total number of non-face card out of 52 cards = 52 - 12 = 40 Therefore, probability of getting ‘a non-face card’ Number of favorable outcomes = 40/52 (vii) a black face card: Cards of Spades and Clubs are black cards. Number of face card in spades (king, queen and jack or knaves) = 3 Number of face card in clubs (king, queen and jack or knaves) = 3 Therefore, total number of black face card out of 52 cards = 3 + 3 = 6 Therefore, probability of getting ‘a black face card’
Number of favorable outcomes = 6/52 (viii) a black card: Cards of spades and clubs are black cards. Number of spades = 13 Number of clubs = 13 Therefore, total number of black card out of 52 cards = 13 + 13 = 26 Therefore, probability of getting ‘a black card’
Number of favorable outcomes = 26/52 (ix) a non-ace: Number of ace cards in each of four suits namely spades, hearts, diamonds and clubs = 1 Therefore, total number of ace cards out of 52 cards = 4 Thus, total number of non-ace cards out of 52 cards = 52 - 4 = 48 Therefore, probability of getting ‘a non-ace’ Number of favorable outcomes = 48/52 (x) non-face card of black colour: Cards of spades and clubs are black cards. Number of spades = 13 Number of clubs = 13 Therefore, total number of black card out of 52 cards = 13 + 13 = 26 Number of face cards in each suits namely spades and clubs = 3 + 3 = 6 Therefore, total number of non-face card of black colour out of 52 cards = 26 - 6 = 20 Therefore, probability of getting ‘non-face card of black colour’ Number of favorable outcomes
= 20/52 (xi) neither a spade nor a jack Number of spades = 13 Total number of non-spades out of 52 cards = 52 - 13 = 39 Number of jack out of 52 cards = 4 Number of jack in each of three suits namely hearts, diamonds and clubs = 3 [Since, 1 jack is already included in the 13 spades so, here we will take number of jacks is 3] Neither a spade nor a jack = 39 - 3 = 36 Therefore, probability of getting ‘neither a spade nor a jack’ Number of favorable outcomes = 36/52 (xii) neither a heart nor a red king Number of hearts = 13 Total number of non-hearts out of 52 cards = 52 - 13 = 39 Therefore, spades, clubs and diamonds are the 39 cards. Cards of hearts and diamonds are red cards. Number of red kings in red cards = 2 Therefore, neither a heart nor a red king = 39 - 1 = 38 [Since, 1 red king is already included in the 13 hearts so, here we will take number of red kings is 1] Therefore, probability of getting ‘neither a heart nor a red king’ Number of favorable outcomes = 38/52 2. A card is drawn at random from a
well-shuffled pack of cards numbered 1 to 20. Find the probability of (i) getting a number less than 7 (ii) getting a number divisible by 3. Solution: (i) Total number of possible outcomes = 20 ( since there are cards numbered 1, 2, 3, ..., 20). Number of favourable outcomes for the event E = number of cards showing less than 7 = 6 (namely 1, 2, 3, 4, 5, 6). So, P(E) = \(\frac{\textrm{Number of Favourable Outcomes for the Event E}}{\textrm{Total Number of Possible Outcomes}}\) = \(\frac{6}{20}\) = \(\frac{3}{10}\). (ii) Total number of possible outcomes = 20. Number of favourable outcomes for the event F = number of cards showing a number divisible by 3 = 6 (namely 3, 6, 9, 12, 15, 18). So, P(F) = \(\frac{\textrm{Number of Favourable Outcomes for the Event F}}{\textrm{Total Number of Possible Outcomes}}\) = \(\frac{6}{20}\) = \(\frac{3}{10}\). 3. A card is drawn at random from a pack of 52 playing cards. Find the probability that the card drawn is (i) a king (ii) neither a queen nor a jack. Solution: Total number of possible outcomes = 52 (As there are 52 different cards). (i) Number of favourable outcomes for the event E = number of kings in the pack = 4. So, by definition, P(E) = \(\frac{4}{52}\) = \(\frac{1}{13}\). (ii) Number of favourable outcomes for the event F = number of cards which are neither a queen nor a jack = 52 - 4 - 4, [Since there are 4 queens and 4 jacks]. = 44 Therefore, by definition, P(F) = \(\frac{44}{52}\) = \(\frac{11}{13}\). These are the basic problems on probability with playing cards. Probability Probability Random Experiments Experimental Probability Events in
Probability Empirical Probability Coin Toss Probability Probability of Tossing Two Coins Probability of Tossing Three Coins Complimentary Events Mutually Exclusive Events Mutually Non-Exclusive Events Conditional Probability Theoretical Probability Odds and Probability Playing Cards Probability Probability and Playing Cards Probability for Rolling Two Dice Solved Probability Problems Probability for Rolling Three Dice 9th Grade Math From Playing Cards Probability to HOME PAGE Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. What is the probability of drawing a black spade out of a standard deck of 52 cards?Hence for drawing a card from a deck, each outcome has probability 1/52. The probability of an event is the sum of the probabilities of the outcomes in the event, hence the probability of drawing a spade is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13.
What is the probability of randomly selecting a spade from a standard 52 card deck?The probability of getting a spade, P(Spade), is 13/52 or 0.2500.
How many black spades are in a deck of cards?26 red and 26 black cards are present in a deck of 52 cards, with 13 spades(black), 13 clubs(black) and 13 hearts(red), 13 diamonds(red)
What is the probability of getting an ace of spades in a deck of 52 cards?The probability of drawing the ace of spades from a deck of cards is 1/52. Probabilities for more than one event can be calculated.
|
