There are four queens in a deck of cards, what is the probability of picking a king or a queen

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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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Mutually Exclusive: can't happen at the same time.

Examples:

• Turning left and turning right are Mutually Exclusive (you can't do both at the same time)
• Tossing a coin: Heads and Tails are Mutually Exclusive
• Cards: Kings and Aces are Mutually Exclusive

What is not Mutually Exclusive:

• Turning left and scratching your head can happen at the same time
• Kings and Hearts, because we can have a King of Hearts!

Like here:

Aces and Kings are Mutually Exclusive (can't be both)		Hearts and Kings are not Mutually Exclusive (can be both)

Probability

Let's look at the probabilities of Mutually Exclusive events. But first, a definition:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Number of ways it can happen: 4 (there are 4 Kings)

Total number of outcomes: 52 (there are 52 cards in total)

So the probability = 4 52 = 1 13

Mutually Exclusive

When two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together:

P(A and B) = 0

"The probability of A and B together equals 0 (impossible)"

A card cannot be a King AND a Queen at the same time!

• The probability of a King and a Queen is 0 (Impossible)

But, for Mutually Exclusive events, the probability of A or B is the sum of the individual probabilities:

P(A or B) = P(A) + P(B)

"The probability of A or B equals the probability of A plus the probability of B"

In a Deck of 52 Cards:

• the probability of a King is 1/13, so P(King)=1/13
• the probability of a Queen is also 1/13, so P(Queen)=1/13

When we combine those two Events:

• The probability of a King or a Queen is (1/13) + (1/13) = 2/13

Which is written like this:

P(King or Queen) = (1/13) + (1/13) = 2/13

So, we have:

• P(King and Queen) = 0
• P(King or Queen) = (1/13) + (1/13) = 2/13

Special Notation

Instead of "and" you will often see the symbol ∩ (which is the "Intersection" symbol used in Venn Diagrams)

Instead of "or" you will often see the symbol ∪ (the "Union" symbol)

So we can also write:

• P(King ∩ Queen) = 0
• P(King ∪ Queen) = (1/13) + (1/13) = 2/13

Example: Scoring Goals

If the probability of:

• scoring no goals (Event "A") is 20%
• scoring exactly 1 goal (Event "B") is 15%

Then:

• The probability of scoring no goals and 1 goal is 0 (Impossible)
• The probability of scoring no goals or 1 goal is 20% + 15% = 35%

Which is written:

P(A ∩ B) = 0

P(A ∪ B) = 20% + 15% = 35%

Remembering

To help you remember, think:

"Or has more ... than And"

Also ∪ is like a cup which holds more than ∩

Not Mutually Exclusive

Now let's see what happens when events are not Mutually Exclusive.

Example: Hearts and Kings

Hearts and Kings together is only the King of Hearts:

But Hearts or Kings is:

• all the Hearts (13 of them)
• all the Kings (4 of them)

But that counts the King of Hearts twice!

So we correct our answer, by subtracting the extra "and" part:

16 Cards = 13 Hearts + 4 Kings − the 1 extra King of Hearts

Count them to make sure this works!

As a formula this is:

P(A or B) = P(A) + P(B) − P(A and B)

"The probability of A or B equals the probability of A plus the probability of B
minus the probability of A and B"

Here is the same formula, but using ∪ and ∩:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

A Final Example

16 people study French, 21 study Spanish and there are 30 altogether. Work out the probabilities!

This is definitely a case of not Mutually Exclusive (you can study French AND Spanish).

Let's say b is how many study both languages:

• people studying French Only must be 16-b
• people studying Spanish Only must be 21-b

And we get:

And we know there are 30 people, so:

(16−b) + b + (21−b) = 30

37 − b = 30

b = 7

And we can put in the correct numbers:

So we know all this now:

• P(French) = 16/30
  
• P(Spanish) = 21/30
  
• P(French Only) = 9/30
  
• P(Spanish Only) = 14/30
  
• P(French or Spanish) = 30/30 = 1
  
• P(French and Spanish) = 7/30

Lastly, let's check with our formula:

P(A or B) = P(A) + P(B) − P(A and B)

Put the values in:

30/30 = 16/30 + 21/30 − 7/30

Yes, it works!

Summary:

Mutually Exclusive

• A and B together is impossible: P(A and B) = 0
• A or B is the sum of A and B: P(A or B) = P(A) + P(B)
  

Not Mutually Exclusive

• A or B is the sum of A and B minus A and B: P(A or B) = P(A) + P(B) − P(A and B)

• And is ∩ (the "Intersection" symbol)
• Or is ∪ (the "Union" symbol)

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