~ 8.4 ~

Estimating Probabilities Through Repeated Experiments

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Learning Targets

I can estimate the probability of an event based on the results from repeating an experiment.
I can explain whether certain results from repeated experiments would be surprising or not.

Notes

A probability for 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 ½, 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.

Experimental probability (practice) | Khan AcademyPractice making reasonable estimates of the likelihood of future events based on past experience.

Making predictions with probability (practice) | Khan AcademyPractice predicting the number of times a certain event will happen.

Probability models (practice) | Khan AcademyPractice creating probability models and understand what makes a valid probability model.

Activities

4.2 In The Long Run

Mai is playing a game where she will win only 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 or show 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?

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.

If the roll stops on one of your winning numbers, what happens in the table?
What appears to be happening with the points on the graph?
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 to fill in the table and graph the results, for a total of 20 points on the graph.
After 20 rolls, what fraction of the total rolls were a win?
How close is this fraction to the probability that Mai will win?

7.8.4.2 In the Long Run7.8.4.2 In the Long Run

Add to Your Notes

A probability tells you how likely an event is to occur.

It is not guaranteed to be an exact match.
If the experiment is repeated many times, the fraction should get close to the calculated probability.
The more repetitions, the more accurate the estimate!

4.3 Due For A Win

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 ⅓. If you flip the coin one more time, will it land heads up to make the cumulative fraction 2/4?

Add to Your Notes

Probability represents the expected likelihood of an event occurring for a single trial on an experiment.

Regardless of previous tries, each coin flip should still be equally likely to land heads up as tails up.
A basketball player who tends to make 75% of his free throw shots will probably make ¾ free throws he attempts, but there is no guarantee that he will make any individual shot, even if he has missed a few in a row.

You conduct a chance experiment many times and record the outcomes. How are these outcomes related to the probability of a certain event occurring?

The fraction of times the event occurs after many repetitions should be fairly close to the expected probability of the event.

What is the probability of rolling a 2, 3, or 4 on a standard number cube?

The probability is ½, 0.5, or 50% since 3 outcomes out of 6 possible are in the even.

You want to roll a 2, 3, or 4 on a standard number cube. If you roll 3 times and none of them result in a 2, 3, or 4, does the probability of getting one of those numbers change with the next roll?

The probability does not change after 3 times.

The probability of getting the flu during flu season is ⅛. If a family has 8 people living in the same house, is it guaranteed that one of them will get the flu?

No, it’s very possible that none of the people will get the flu, and also possible that more than 1 person will get the flu.

The probability of getting the flu during flu season is ⅛. If a country has 8 million people, about how many do you expect will get the flu? Does this number have to be exact?

We might expect about 1 million people in the country to get the flu, but this is probably not exact.

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