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Card Game Probability

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In this article, we will learn how to calculate probabilities when a card is drawn from a pack of standard 52 playing cards. In order to calculate card game probabilities, we should first understand what kinds of cards are used in the card game.

  1. The entire pack of cards has 52 cards in total.
  2. The playing cards are divided into four groups (called “suits”) – 13 spades, 13 clubs, 13 hearts and 13 diamonds.
  3. Each group of 13 cards consists of a king, a queen, a jack, an ace and the number cards 2, 3, .., 9.
  4. The king, queen and jack cards are called the face cards. Since there are 3 face cards in each of the four groups there are 12 face cards in all.
  5. Diamond and hearts are red cards, so there are 26 red cards in all.
  6. Spades and clubs are black cards, so there are 26 black cards in all.
Card Game probability

We can calculate probabilities using the formula,

Required Probability = Number of favourable outcomes/ Number of total outcomes

Examples of calculating card game probability:

Problem 1: A card is drawn from a pack of 52 playing cards. Find the probability that the card chosen is:

  1. A black card
  2. A face card
  3. A red face card
  4. A black ace

Solution:

a) There are 26 black cards and so we have 26 favourable outcomes from a total of 52 possible outcomes.

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 26/52 = ½ = 0.5

b) There are 12 face cards and so we have 12 favourable outcomes from a total of 52 possible outcomes.

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 12/52 = 3/13 = 0.2307

c) There are 3 face cards in the suit of hearts and 3 face cards in the suit of diamonds.  So we have 6 red face cards in all.

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 6/52 = 3/26 = 0.1153

d) There is one face card in the suit of spades and one ace card in the suit of spears.  So we have 2 black ace cards in all.

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 2/52 = 1/26 = 0.03846

Problem 2: A card is drawn from a pack of 52 playing cards. Find the probability that the card chosen is:

  1. A card having the number 2 or 3
  2. A card having even number.

Solution:

a) There is one number 2 card in each of the four suits and one number 3 card in each of the four suits so we have 8 favourable outcomes.

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 8/52 = 2/13 = 0.1538

b) The even number cards are 2, 4, 6, and 8. There are four cards of each number and so we have 16 favourable outcomes

Therefore, Required Probability = Number of favourable outcomes/ Number of total outcomes = 16/52 = 4/13 = 0.3076

Basic Strategy to Solve Problems relating to Card Game Probability:

1. Make sure that you have memorized the four types of different playing cards and their associated colours.

2. Given a particular question, break the given condition into two or more simpler parts and then calculate the number of favourable outcomes. For example, if you are asked to find the probability of getting a red king we analyze the situation as follows. There are two conditions to look out for – Red and King. We know that there are 26 red cards and 4 kings. How many of the kings are red? The answer presents itself easily if we analyse the question in a systematic manner – King of Hearts and King of Diamonds. Thus there are two favourable outcomes.

3. Divide the number of favourable outcomes by 52 (since there are 52 playing cards), in order to obtain the desired probability.

4. This basic strategy is very helpful in solving all kinds of probability related problems.

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