What is the probability of picking a king queen Jack or an Ace of diamonds in 52 playing cards?

Probability can be defined as a chance of occurring a particular event. The range of probabilities lies between 0 to 1, or 0% to 100% in terms of percentage. If the probability of an event is 0, the event is unlikely to happen and considered an impossible event. On the other hand, if the probability of an event is 1, it will be a certain event. In a sample space, The probabilities of all the events add up to 1. Some application of probability is in weather forecasting, flipping a coin, rolling dice, in sports and board games.

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Important terms related to probability
Similar Problems
Summary of Basic Probability

Before learning about the terms used in probability. Let’s first take a look at the basic definition of probability and the formula used to find the probability. Probability basic tells how likely an event will occur. The formula for probability is,

Probability of an event = Number of favorable outcomes / Total number of outcomes

Note The probability always lies between 0 and 1.

0 ≤ P ≤ 1

For example:

Probability of getting a head on a fair two-faced coin = 1/2

As here each coin has 1 head and the total outcome is 2.

Experiment: An observation whose result is not yet known is called an experiment.
Random Experiment: It is an observation that is repeated a number of times and results in different outcomes. Predicting the result of a random experiment is called a probability. Drawing a card from a deck is an example of a random experiment as on each turn, you will get a different card.
Outcome: Random experiments yield different results, known as outcomes. Suppose if we flip a coin and we get head. So, tossing a coin is a random experiment that yielded the result “head”.
Sample space: All the possible outcomes of a random experiment constitute sample space. For example, if we roll a die, we can get 1,2,3,4,5, or 6. As a result, the sample space will have 1, 2, 3, 4, 5, and 6. This means that if a die is rolled, the sample space or the possible outcomes is 6.
Event: When a single experiment occurs, its result is called an event. Getting a Head when tossing a coin is an example of an event. It is generally denoted by “E”.
Possible Outcomes: All the outcomes in an experiment that is likely to occur are possible outcomes. For example, on tossing a coin, we either get head or tails, so there are 2 possible outcomes in this situation.
Impossible Event: An event whose possibility of happening is 0 is called an impossible event. For example, getting a 17 on rolling a 6 faced-dice is impossible.
Independent Events: Two events are said to be independent if the occurrence of one event does not affect the occurrence of the second event and vice-versa

Solution:

A deck of 52 cards consists of 4 suits: diamonds, hearts, clubs, and spades.

There are 13 cards in a suit, they are: Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2.

Each suit has only one Queen card.

Therefore, In a deck of 52 cards, there is only one queen of Diamonds.

Summary of Basic Probability

The 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.

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