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Statistics Empirical and Theoretical Probability

Question 1

In quantum mechanics, the photoelectric effect occurs when a sufficient-energy photon strikes a metal, and the metal emits an electron.(Albert Einstein got the Nobel Prize for figuring this out!).Suppose we conduct this experiment by shining a light source at a metal, and then counting electrons emitted per second using a small detector (so we’re actually only counting a small portion of the electrons emitted, but let’s just call what we detect the “emitted” electrons to make things simple).

1. Let us suppose that we count 4 electrons emitted in a period of 10 seconds. From this data what is the empirical probability of an electron being emitted in a given second?

2. We somehow calculate that the theoretical probability of an electron being emitted in a given second is P(emitted)=0.5. What is the probability of an electron not being emitted in a given second?

3. Suppose we sample for emitted electrons for 3 seconds (so in each of the 3 seconds, an electron is either emitted or not emitted as per the probabilities from questions number 2). What is the sample space for this experiment?

4. Based on the sample space from question number 3, what is the probability of 2 electrons being emitted during the 3 seconds of our experiment?

5. Using P(emitted)=0.5 for a given second, use the multiplication rule to calculate the probability of an electron being emitted in each of the 3 seconds (so 3 electrons are sequentially emitted). How does this compare with the probability of this event calculated by looking at the sample space?

6. The multiplication rule allow you to determine that the probability of an electron being emitted in both the first and second seconds (that’s not a typo) is 0.25. Use the addition rule to determine the probability of an electron being emitted in the first second, or the second second, or both.

7. Now suppose we change our experiment, and first use light source “A” for 100 seconds, and then use light source “B” for 80 seconds. For light source A an electron is emitted in 52 of the 100 seconds, and for light source B an electron is emitted in 36 of the 80 seconds. Make a table showing this data.

8. What is the probability of an electron being emitted in a given second using light source A?

9. What is the probability of an electron being emitted in a given second using light source B?

10. What is the probability of an electron being emitted in a given second over the whole course of the experiment?

11. What is the probability that in a given second of our experiment we used light source B or an electron was not emitted?

12. Suppose we are assembling an engineered molecule, and we have 10 different sub-units we can string together. If we only use 5 sub-units, how many different engineered molecules can we make? (Order counts, and this is “without replacement,” so we start with only one of each of the sub-units.)

13. Now suppose order doesn’t count (which is not true, but suppose we’re not very good chemists and we don’t know this). How many different engineered molecules do we think we can make, using 5 of the 10 sub-units?

14. Now suppose we have a large supply of each of the sub-units (so this can be considered “with replacement.” Calculate the answer as per question #12 but “with replacement.” Hint: You can use the multiplication counting rule.

15. Explain why your answers to questions #12, #13, and #14 are different.