Unit 7 Probability and Counting

Reed Halpert and Quintin Campbell

Probability and Counting

Probability is the study of how likely an event is to happen. We can use many different methods to do this and it applies to a variety of real world situations. A game of poker, being struck by lightning, or being eaten by a shark are all events that we can calculate the likelihood of happening. This page is going to cover all of the material gone over in the probability unit to prepare you for the final exam.

Diagrams and Charts

There are three ways to visually represent a probability problem: Tree diagrams, area diagrams, and a decision chart.

Tree Diagram

Here we see a tree diagram detailing a one and one free throw for a 70% free throw shooter. We take the first free throw and write out the chances of each result. There is a 70% chance he makes it, 30% chance he misses it. If he makes it, he earns a second free throw, in which he has the same chances of making and missing it. To calculate the probabilities of the three outcomes, we multiply the percentages of a make or miss. For two points, we multiply the 70% make percentage twice to get a .49, or 49% chance of making both free throws. The pattern follows for the other scenarios.

Area Diagram

The Area Diagram follows a similar pattern as the tree diagram, the only difference being it is drawn differently. We are using the same situation for this diagram. The first free throw is represented on the left side and the second is written on the top. To calculate the chances of an event happening, simply multiply the percentages that align with a certain box. To figure out the chance of making one free throw and missing the next, multiply .7 by .3 to get .21, or 21% chance of that happening.

Decision Chart

The Decision Chart requires a different situation than the one and one free throw. Each charts have their own limitations, which are discussed below. The scenario for our decision chart is picking 5 random playing cards from a group of 25. Because no playing card is the same, when you take one away, you have one less option to choose from. The chart shows how many choices you have as you take the cards. When picking your 4th card, you have 22 options to choose from. To calculate the number of possibilities, multiply the 5 amounts of choices together. This rule applies to a decision chart with 2 decisions or X decisions. After calculating the number of possibilities, you can now calculate the probability of a scenario. We have named 5 random cards (assuming they are apart of the 25) and written them down. That is 1 possible scenario out of 6,375,600. To calculate the chance of that sequence, divide 1 by 6,375,600.

Independent Events

An independent event is one where the way the outcome occurs for one event does affect the probability of the outcome for a second event. Events like these are a coin flip or drawing a playing card from a deck.

Mutual Exclusivity

Mutual Exclusivity means that the outcome of one event makes it impossible for the outcome of another event to occur. For instance, rolling two die and taking the sum of them. It is impossible for the sum to be both odd and even.

It is important to determine if the event you are calculating the probability for is mutually exclusive or independent because it affects the operation we will perform.

Combinations

A combination is a method of examining arrangements. One example we can use is a hand in 5 card poker. A hand you can receive is 3 of hearts, 5 of spades, Jack of Clubs, King of Spades, and 2 of Diamonds. You can also receive the hand 5 of spades, Jack of clubs, 3 of hearts, 2 of Diamonds, and King of Spades. Notice that these hands include the same exact cards. They are the same hand. This is a perfect example of a combination.

In Combinations, order does not matter, and depending on the circumstance of the situation, items can repeat. The 5 card poker hand cannot repeat simply because there is no such thing as two of the same card. We are choosing 5 cards out of 52. However, there are situations where it can repeat, so it is important to keep that in mind.

We can do a combination on our calculator.

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How to do a Combination on the calculator

This is done using our 5 card poker hand. The end result is the number of possibilities or the number of possible hands. In order to find the probability of getting our model hand, we do 1 divided by the number of possibilities, and then turn that into a percentage.

Click on the video and then click on the square with the arrow in the top left if the video does not load.

Permutations

A permutation is a method for counting arrangements. We can't use our 5 card poker hand because order matters and there are no repeats. An election is a good representation of a permutation. If Timmy, Billy, Willy, Kyle, and Nathan are all running for 3 student council spots, it matters who you pick and in what order. If you want Willy to be president, you put him first, not third. In this case we are picking and placing 3 out of 5. You also cannot vote for Kyle all 3 times.

We can do a permutation on our calculator.

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How to do a Permutation on the calculator

Here we will calculate the number of possibilities to pick and place 3 out 5 members in our election. To find the probability of any number of scenarios, divide your desired number of scenarios by the total possibilities shown on the calculator.

Click on the video and then click on the square with the arrow in the top left if the video does not load.

Applications

We use probability everyday. Whether it's deciding which lottery ticket to buy in order to win the lottery or randomly get chosen to represent your form in EA week festivities. However, one specific real life application of probability is in a game of cards. Every card game revolves around the premise of probability. In a game of blackjack, for example, you have to know the probability of drawing a card that will keep you under 21. That being said, a smart player will know that the smaller the number in his hand is, the higher the probability that the card that he draws will keep him in play.

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Example Problem

This is how to calculate the possibility of getting a full house in 5 card poker.

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