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Table of Contents
In this lesson, we will develop a probability model and use it to find probabilities of events. Moreover, we will also compare probabilities from a model to observed frequencies, and even find probabilities of compound events using organized lists, tables, and tree diagrams.
See the fact file below for more information on the working with probability models or alternatively, you can download our 28-page Working with Probability Models worksheet pack to utilise within the classroom or home environment.
Key Facts & Information
INTRODUCTION TO PROBABILITY MODELS
- A formal definition of probability starts with a sample space S, which is any set that lists all possible outcomes of an unknown experiment or situation.
- S = {rain, snow, clear}
- The above example might be used for predicting tomorrow’s weather; or perhaps S is the set of all positive real numbers, when predicting next week’s stock price.
- S can be any set at all, even an infinite set. We usually write s for an element of S, so that means s ∈
- Note that S includes only those elements that we are interested in.
- A probability model consists of a nonempty set called the sample space S; a collection of events that are subsets of S; and a probability measure P assigning a probability between 0 and 1 to each event, with P(∅) = 0 and P(S) = 1.
- For the weather example, the subsets {rain}, {snow}, {rain,snow},{rain, clear}, {rain, snow, clear}, and even the empty set ∅ = { }, are all examples of subsets of S that could be events.
- The comma used in these subsets represents “or”; thus {rain, snow} is the phenomenon that it will rain or snow. We will generally assume that all subsets of S are events.
- A probability model needs a probability measure P. The probability measure must assign, to each event A, a probability P(A) having the following properties:
- P(A) is always a nonnegative real number, between 0 and 1 inclusive
- P(∅) = 0, such that if A is the empty set ∅, then P(A) = 0
- P(S) = 1, such that if A is the entire sample space S, then P(A) = 1
- P is (countably) additive, meaning that if A1, A2, … is a finite sequence of disjoint events, then:
- P(A1 ∪ A2 ∪ … ) = P(A1) + P(A2) + …
- Given S = {rain, snow, clear}, the probability of rain is 40%, snow is 15%, and a clear day is 45%.
- We can write this as P({rain}) = 0.40, P({snow}) = 0.15, and P = ({clear}) = 0.45.
- Obviously, P(∅) = 0 since it is impossible that nothing will happen in the weather the next day. Also P({rain, snow, clear}) = 1 because there will be exactly one of rain, snow, or clear that must happen tomorrow.
- Now, what is the probability that it will rain or snow tomorrow? Using the additive property, we can say that:
- P({rain, snow}) = P({rain}) + P({snow}) = 0.40 + 0.15 = 0.55
- Therefore, there is a 55% chance of rain or snow tomorrow.
- Suppose we flip a fair coin, which can come up with either heads (H) or tails (T) with equal probability.
- S = {H, T}, with P(H) = P(T) = 0.5
- To check, P(H) + P(T) = 1
- Suppose we flip three fair coins in a row and keep track of the sequence of heads and tails that result.
- S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
- Each of the eight outcomes is equally likely. Thus, P(HHH) = ⅛, P(TTT) = ⅛, and so on. Also, the probability that the first coin is heads and the second coin is tails, but the third coin can be any of the two, is equal to the sum of the probabilities of the events HTH and HTT, such that P(HTH) + P(HTT) = ⅛ + ⅛ = ¼.
PROBABILITY OF COMPOUND EVENTS
- If an event has only one possible outcome, it is called a simple (or single) event. Otherwise, the event is called a compound event.
- A tree diagram is a drawing with “line segments” that illustrates sequentially the possible outcomes of a given event.
- Look at the tree diagram for the toss of a coin.
- You are off to soccer, and love being the goalkeeper, but that depends who is the coach today: (1) with coach Sam, the probability of being goalkeeper is 0.5; and (2) with coach Alex, the probability of being goalkeeper is 0.3.
- Coach Sam is usually available, about 6 out of every 10 games (a probability of 0.6). So, what is the probability you will be a goalkeeper today?
- Show the two possible coaches: Sam or Alex.The probability of getting Sam is 0.6, so the probability of getting Alex must be 0.4. Altogether, the probability is 1.
- Now, if you get Sam, there is a 0.5 probability of being goalkeeper (and 0.5 of not being goalkeeper):
- If you get Alex, there is a 0.3 possibility of being goalkeeper (and 0.7 of not being goalkeeper).
- Since the tree diagram is now complete, let’s solve for the overall possibilities. This is done by multiplying each probability along the “branches” of the tree.
- On the right is how you compute for the “Sam, Yes” branch.
- When Alex is coach, we get these results:
- A 0.4 chance of Alex as coach, followed by a 0.3 chance gives 0.12 possibility. Now, we add the column:
- 3 + 0.12 = 0.42 (42% chance) probability of being goalkeeper today.
- Using the organized list method, you would list all the different possible outcomes that could occur. This can be difficult because there’s a high probability that we will forget one or two options.
- If you flip a coin and roll a die, what is the probability of getting tails and an even number?
- First, we need to start by listing all the possible outcomes we could get. Take note that H1 means flipping a head and rolling a 1.
- S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4. T5, T6}
- There are 12 possible outcomes, and three of these outcomes give the desired outcome (tail plus an even number). These are T2, T4, and T6.
- Therefore, the probability is: P = 3/12 = 1/4 =25%
- If you flip a coin three times, what is the probability of flipping at least 2 heads?
- Now, we are working with three different events – each flip counts as an individual event.
- Flip 1 H H H H T T T T
- Flip 2 H H T T H H T T
- Flip 3 H T H T H T H T
- There are 8 different outcomes. These are the favorable ones: HHT, HTH, THH, and HHH (at least 2 heads includes flipping three). So the probability is: P = 4/8 = 50%
- Area models can be used to represent simple probabilities. The whole figure represents the total number of possible outcomes. The shaded part represents the desired outcomes.
- If you flip a coin and roll a die, what is the probability of getting tails and an even number?
- Start by creating a table with the outcomes of one event listed on the top and the outcomes of the second event listed on the side. Fill in the cells of the table with the corresponding outcomes for each event. Shade the cells that fit the given probability.
- There are 12 cells, of which three are shaded. So the probability is: P = 3/12 = 1/4 = 25%
Working with Probability Models Worksheets
This is a fantastic bundle which includes everything you need to know about the working with probability models across 28 in-depth pages. These are ready-to-use Working with Probability Models worksheets that are perfect for teaching students how to develop a probability model and use it to find probabilities of events. Moreover, we will also compare probabilities from a model to observed frequencies, and even find probabilities of compound events using organized lists, tables, and tree diagrams.
Complete List Of Included Worksheets
- Lesson Plan
- Working with Probability Models
- Coins In Her Pocket
- Books In a Shelf
- Sum of Six
- Sum of Seven or Nine
- With or Without Repetition
- Clothes In the Closet
- Toss and Flip
- Creating Events
- Taking Chances
- Test Yourself
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Link will appear as Working with Probability Models Facts & Worksheets: https://kidskonnect.com - KidsKonnect, August 18, 2020
Use With Any Curriculum
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