17 Basic Probability
Learning Objectives
- Identity possible probability values
- Create sample spaces for events
- Find basic probability using classical method and Empirical approach
What is a probability? What numbers can be probability values?
Probability is the likelihood of something happening. A probability is a fraction or decimal number between 0 and 1 (inclusive). We denote the probability of an event E, P(E).
Examples: Probabilities
- Identify the numbers below that cannot be a probability.
1.50, 0.53, -0.25, 9/3, 3/4, 1.
Solution: 1.50, -.025, and 9/3 are all not possibly probabilities because they are not between 0 and 1.
2. There is a 36% chance of getting 7 red M&M’s in a fun size bag. What is the probability (as a decimal) of getting 7 red M&M’s in a fun size bag?
Solution: 0.36
Probability Terms and Special Situations
The probability of an impossible event is 0.
Example of an impossible event: Event: rolling a 7 on a six-sided die
The probability of an event that is certain to occur is 1.
Example of an event that is certain to occur: Event: rolling a number between 1 and 6 inclusively on a six-sided die
Experiments, Events, and Sample Space
In probability, an experiment is any process that can be repeated in which the results are unknown. A result of an experiment is called an outcome. The sample space, S, of a probability experiment is the collection of all possible outcomes.
Examples: Samples Spaces
- Consider the experiment of tossing a coin three times. The coin can either land on heads (H) or tails (T).
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- What are possible events for the given experiment?
- Solution: All tosses landing on Heads, two tosses landing on H and one toss on T, one toss landing on H and two tosses landing on T, all tosses landing on T.
- What is the sample space?
- Solution: Sample space includes all possible WAYS that the coins can land.
- S = { HHH, HHT, HTH, THH, HTT, THT, HTT, TTT}
- Notice how the sample space includes the event “two tosses landing on H and one toss on T” three different times- since this can happen in one of three ways (T landing 3rd, 2nd, or 1st).
- What are possible events for the given experiment?
2. Consider the experiment of rolling a 4-sided die and drawing a card from a 52-card deck to determine the suit. For more information about standard 52-card decks see here. These cards come in four different “suits” . They are Hearts (H), Clubs (C), Diamonds (D), and Spades (S).
What is the sample space?
Solution: S = {1H, 2H, 3H, 4H, 1C, 2C, 3C, 4C, 1D, 2D, 3D, 4D, 1S, 2S, 3S, 4S}
Computing Probabilities Using Classical Method
If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E, is
[latex]P(E) = \frac{\text{number of ways E can occur}}{\text{number of possible outcomes}}= \frac{m}{n}[/latex]
Important Note: The classical method of computing probabilities requires equally likely outcomes.
Examples: Computing Probabilities using Classical Method
- Use the information in Example 1.
- Find the probability for each event stated in 3 (a).
- Solution: P(HHH)= 1/8, P(TTT)= 1/8, P(2 H and 1 T)= 3/8, P(1 H and 2T) = 3/8.
- Notice how the probability of 2H and 1 T is three times the probability of HHH, since this event (2 H and 1T) happens three times as often in the sample space for this experiment.
- Solution: P(HHH)= 1/8, P(TTT)= 1/8, P(2 H and 1 T)= 3/8, P(1 H and 2T) = 3/8.
- Find the probability for each event stated in 3 (a).
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- What is the probability of getting at least one heads?
- Solution: At least one H means that we could have one, two, OR three heads, but one is the smallest amount we can have (no fewer!). This happens in all of the situations except when we have tossed three tails. So this means it happens in the situations { HHH, HHT, HTH, THH, HTT, THT, HTT} There are 7 ways this happens, 8 total ways the three coins can land, so the probability is of at least one heads is 7/8.
- What is the probability of getting at least one heads?
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- What is the probability of getting at most one heads?
- Solution: At most one H means that we can get one H but NO MORE! So this means we could have one H or zero H. This happens in the situations { HTT, THT, HTT, TTT}. There are four situations here and eight total ways the coin can be tossed three times, so the probability of at most one heads is 4/8= 1/2 or 50%.
- What is the probability of getting at most one heads?
Approximating Probabilities Using the Empirical Approach
The probability of an event E is approximately the number of times event E is observed divided by the number of trials of the experiment.
[latex]P(E) \approx \frac{\text{frequency of E}}{\text{number of trials of the experiement}}[/latex]
Examples: Computing probabilities using the empirical approach.
- Consider the following experiment: Six cards are numbered from 1 to 6 without repetition. One card is drawn at a time and the number is recorded and then the card is placed back in the deck. This is done 15 times and the results from the experiment are represented in the table below.
|
Outcomes |
Frequency |
|
1 |
3 |
|
2 |
0 |
|
3 |
5 |
|
4 |
3 |
|
5 |
1 |
|
6 |
3 |
- Based on these results, what is the probability of getting a 3?
- Solution: This experiment is performed 15 total times. The outcome of 3 appears five times. This means the probability of getting a three is 5/15 = 1/3.
- Based on these results, what is the probability of getting a number between 1 and 6 (inclusive)?
- Solution: This experiment is performed 15 total times. The only possible options for drawing a card are those numbers between 1 and 6 inclusive. Notice also that the sum total of the frequecies is also 15. So the probability of getting a number 1 to 60 is 15/15 = 1 or 100%. This means, not surprisingly that you will for sure draw a number between 1 and 6.
2. The procedure was repeated 15 more times and the results were recorded in the table below.
|
Outcomes |
Frequency |
|
1 |
1 |
|
2 |
3 |
|
3 |
3 |
|
4 |
2 |
|
5 |
2 |
|
6 |
4 |
- Based on these new results, what is the probability of getting a 3?
- Solution: Based on these 15 draws, 3 appears three times. So the probability of a 3 is 3/15=1/5 or 0.20 or 20%.
- If one card is randomly selected from the deck, what is the probability of getting a 3?
- Solution: Considering both records of cards, we see that the 3 appeared 5+3 = 8 total times out of 15+15= 30 total draws. This means we can better approximate the probability of getting a 3 as 8/30.
The Law of Large Numbers
The law of large number states as the number of trails increase, the empirical approach for the experiment approaches the classical probability or the true probability.
This “law” should seem fairly obvious… the more data that is collected or the more times the experiment is carried out- the better using results of the experiment are to predict the true probability.
Examples: Law of Large Numbers
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- If a coin is tossed once, what is the probability of getting heads with the Classical Method?
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- Solution: Since the outcome of tossing a coin is Heads or Tails (and each are the same likely to occur), the probabiliyt of tossing a heads in one toss is 1/2 or 50%.
2. Suppose you carried out several trials of tossing a coin. For each trial, you recorded the number of coins tossed, and the number of tails and heads that appeared. You Find the probability of getting heads based on the results of this experiment. Please work out these probabilities on your own before seeing the results in the solutions table.
Trial
Number of Coin Tosses
Number of Tails
Number of Heads
Probability of getting heads based on the results of the experiment. (Round answer to three decimal places)
Trial 1
1
0
1
1/1
Trial 2
25
17
8
Trial 3
50
20
30
Trial 4
200
97
103
Trial 5
500
248
252
Solutions:
Trial
Probability of getting heads based on the results of the experiment. (Round answer to three decimal places) Trial 1
1/1 = 1 Trial 2
8/25 Trial 3
30/50 Trial 4
103/200 Trial 5
252/500 What relationship do you notice about the number of coin tosses and the probability of each experiment?
-
Solution: As the number of times the times the experiment is repeating, the probability for the experiment or the empirical approach trends toward the classical probability for heads.
Unlikely versus Likely
An event is unlikely or unusual if its probability is very small, such as 0.05 (5 %) or less. Otherwise, the probability is considered likely or not unusual or usual.
Examples
Example:
- A survey was given to 350 randomly selected UT students. The survey showed that 218 students attended a home football game during the 2022 season. What is the probability that a randomly selected UT student attended a home football game during the 2022 season? Round to 3 decimal places.
Interpret this probability. If 1000 UT students were surveyed, we would expect about
Would it be unusual for a UT student to attend a home football game during the 2022 season? Explain.
Content created in 2022 by the RSCC Math 1410/1420 OER Team. This work is licensed under a Creative Commons Attribution 4.0 International License
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