8.1.4: Estimating Probabilities Through Repeated Experiments
Lesson
Let's do some experimenting.
Exercise \(\PageIndex{1}\): Decimals on the Number Line
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Locate and label these numbers on the number line.
- \(0.5\)
- \(0.75\)
- \(0.33\)
- \(0.67\)
- \(0.25\)
- Choose one of the numbers from the previous question. Describe a game in which that number represents your probability of winning.
Exercise \(\PageIndex{2}\): In the Long Run
Mai plays a game in which she only wins if she rolls a 1 or a 2 with a standard number cube.
- List the outcomes in the sample space for rolling the number cube.
- What is the probability Mai will win the game? Explain your reasoning.
- If Mai is given the option to flip a coin and win if it comes up heads, is that a better option for her to win?
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Begin by dragging the gray bar below the toolbar down the screen until you see the table in the top window and the graph in the bottom window. This applet displays a random number from 1 to 6, like a number cube. Mai won with the numbers 1 and 2, but you can choose any two numbers from 1 to 6. Record them in the boxes in the center of the applet.
Click the Roll button for 10 rolls and answer the questions. - What appears to be happening with the points on the graph?
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- After 10 rolls, what fraction of the total rolls were a win?
- How close is this fraction to the probability that Mai will win?
- Roll the number cube 10 more times. Record your results in the table and on the graph from earlier.
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- After 20 rolls, what fraction of the total rolls were a win?
- How close is this fraction to the probability that Mai will win?
Exercise \(\PageIndex{3}\): Due For A Win
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For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning.
- You flip the coin once, and it lands heads up.
- You flip the coin twice, and it lands heads up both times.
- You flip the coin 100 times, and it lands heads up all 100 times.
- If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning.
- If you flip the coin 100 times, what are some other results that would not be surprising?
- You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently \(\frac{1}{3}\). If you flip the coin one more time, will it land heads up to make the cumulative fraction \(\frac{2}{4}\)?
Summary
A probabilityfor an event represents the proportion of the time we expect that event to occur in the long run. For example, the probability of a coin landing heads up after a flip is \(\frac{1}{2}\), which means that if we flip a coin many times, we expect that it will land heads up about half of the time.
Even though the probability tells us what we should expect if we flip a coin many times, that doesn't mean we are more likely to get heads if we just got three tails in a row. The chances of getting heads are the same every time we flip the coin, no matter what the outcome was for past flips.
Glossary Entries
Definition: Chance Experiment
A chance experiment is something you can do over and over again, and you don’t know what will happen each time.
For example, each time you spin the spinner, it could land on red, yellow, blue, or green.
Definition: Event
An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes.
Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”
Definition: Outcome
An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails.
Definition: Probability
The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen.
For example, the probability of selecting a moon block at random from this bag is \(\frac{4}{5}\).
Definition: Random
Outcomes of a chance experiment are random if they are all equally likely to happen.
Definition: Sample Space
The sample space is the list of every possible outcome for a chance experiment.
For example, the sample space for tossing two coins is:
| heads-heads | tails-heads |
| heads-tails | tails-tails |
Practice
Exercise \(\PageIndex{4}\)
A carnival game has 160 rubber ducks floating in a pool. The person playing the game takes out one duck and looks at it.
- If there’s a red mark on the bottom of the duck, the person wins a small prize.
- If there’s a blue mark on the bottom of the duck, the person wins a large prize.
- Many ducks do not have a mark.
After 50 people have played the game, only 3 of them have won a small prize, and none of them have won a large prize.
Estimate the number of the 160 ducks that you think have red marks on the bottom. Then estimate the number of ducks you think have blue marks. Explain your reasoning.
Exercise \(\PageIndex{5}\)
Lin wants to know if flipping a quarter really does have a probability of \(\frac{1}{2}\) of landing heads up, so she flips a quarter 10 times. It lands heads up 3 times and tails up 7 times. Has she proven that the probability is not \(\frac{1}{2}\)? Explain your reasoning.
Exercise \(\PageIndex{6}\)
A spinner has four equal sections, with one letter from the word “MATH” in each section.
- You spin the spinner 20 times. About how many times do you expect it will land on A?
- You spin the spinner 80 times. About how many times do you expect it will land on something other than A? Explain your reasoning.
Exercise \(\PageIndex{7}\)
A spinner is spun 40 times for a game. Here is a graph showing the fraction of games that are wins under some conditions.
Estimate the probability of a spin winning this game based on the graph.
Exercise \(\PageIndex{8}\)
Which event is more likely: rolling a standard number cube and getting an even number, or flipping a coin and having it land heads up?
(From Unit 8.1.2)
Exercise \(\PageIndex{9}\)
Noah will select a letter at random from the word “FLUTE.” Lin will select a letter at random from the word “CLARINET.”
Which person is more likely to pick the letter “E?” Explain your reasoning.
(From Unit 8.1.3)