Probability Problem

I am a teacher (social sciences, civics and law), but not a math teacher. I am developing a probability or statistic problem for the class and I need help determining the right result. It involves multiple events and the purpose is to use statistics and probability in proving a case of nuisance. The scenario is below. Please help me determine the probability and if you can clarify it a little for me, please help as well.

I got the result of 100% for the first event (event A) and event B I got 50%. I do not know if this is right or not. It has been a long time since I have performed statistics. But, I figured it is a good exercise for the class.

Scenario: Gary has been out of school for some time. He receives an email stating he registered for class on the school's system, but he did not and someone else used his email address. This email was sent on his birthday. The only way to register was to have access to Gary's school account (username and password). In this event, Gary's information was changed as well.

Three months later, Gary gets another email stating he registered for class but on a different school site (this school will be called the second school). Again, someone has used his email address. But, Gary does not use the second school's site and has not ever used the second school's site. This email was also sent on his wife's birthday.

What is the statistics or probability of this happening?

prob2013 said:

I am a teacher (social sciences, civics and law), but not a math teacher.
I am developing a probability or statistic problem for the class and I need help determining the right
It involves multiple events and the purpose is to use statistics and probability in proving a case of
nuisance. The scenario is below. Please help me determine the probability and if you can clarify it a
little for me, please help as well.

I got the result of 100% for the first event (event A) and event B I got 50%.
I do not know if this is right or not. It has been a long time since I have performed
statistics. But, I figured it is a good exercise for the class.

Scenario: Gary has been out of school for some time. He receives an email stating he registered for
class on the school's system, but he did not and someone else used his email address.
>>>This email was sent on his birthday.<<< The only way to register was to have access to
Gary's school account (username and password). In this event, Gary's information was changed as well.

Three months later, Gary gets another email stating he registered for class but on a different school site
(this school will be called the second school). Again, someone has used his email address. But, Gary does
not use the second school's site and has not ever used the second school's site.
>>>This email was also sent on his wife's birthday.<<<

What is the statistics or probability of this happening?

Click to expand...

The two highlighted statements above contradict each other. The first mentions his birthday,
while the second mentions his wife's birthday. And you used the word "also" in the second
statement.

- - - - - - - - - -- -

I don't see how you can expect answers (read: probabilities for these two scenarios)
for these when there are many contributing factors and of different probability weights.

simplifying the scenario

lookagain said:

The two highlighted statements above contradict each other. The first mentions his birthday,
while the second mentions his wife's birthday. And you used the word "also" in the second
statement.

- - - - - - - - - -- -

I don't see how you can expect answers (read: probabilities for these two scenarios)
for these when there are many contributing factors and of different probability weights.

Click to expand...

Thanks. I think I need to go back and simplify the scenario. But, I think it is a scenario based on the probability of coincidences or rare events. I am trying to show that the coincidence is not a coincidence or random but specific and the couple experienced a intentional nuisance.

One way I guess would be to consider the events as independent. Any thoughts?

prob2013 said:

so, how should I simplify it? for example, what if only the first event is considered? the possible outcomes is that it was a mistake or identity theft occurred? could an answer be generated? if so, what would it be?

Click to expand...

Your scenario states that certain things did happen, but there is no way to get probability from that.

Probability is the ratio of "number of successes" to "total number of trials." [That is also called "empirical" probability because it is the result of a measurement with a finite number of trials.] If you state that an event happened, and we accept your statement, the probability becomes 1 out of 1, or 100% - but there is no statistical information in that.

If on the other hand you say the school database has 10,000 records, and 30 of them were hacked, you could ask for the probability that Gary's record was hacked. The second event is more complex than that - ir would seem to deal with conditional probabilities, such as "Given that event A happened, phat is the probability of event B."

Are you looking for the probability of guessing the birthday? And then the combined probability of guessing two independent birthdays? [For independent events, the probability that both are true is the product of the two probabilities.]

I think it is a great idea to introduce statistics in the civics context. Especially if you get as far as defining what it means when a pollster says "the margin of error is 4%." That is clearly not an "empirical" probability, since they only took the poll once. To get that far, you have to introduce the "normal" distribution (familiar name being the "bell curve"), and the binomial distribution, and finally the sampling theorem. In those cases probability is found as the fractional area under a specific portion of the distribution.

simplifying the scenario

Thanks. All of you are giving me great ideas.

If on the other hand you say the school database has 10,000 records, and 30 of them were hacked, you could ask for the probability that Gary's record was hacked. The second event is more complex than that - ir would seem to deal with conditional probabilities, such as "Given that event A happened, phat is the probability of event B."

Yes, this is what I am looking for. I am looking for the probability that Gary's email was hacked.

Are you looking for the probability of guessing the birthday? Yes.

And then the combined probability of guessing two independent birthdays? Yes.

prob2013 said:

Thanks. I think I need to go back and simplify the scenario. But, I think it is a scenario based on the probability of coincidences or rare events. I am trying to show that the coincidence is not a coincidence or random but specific and the couple experienced a intentional nuisance.

One way I guess would be to consider the events as independent. Any thoughts?

Click to expand...

ok - this is another branch of statistics called "hypothesis testing." You set up what is called a "null hypothesis" and an "alternate hypothesis." The logic goers like this:
H0: events A and B are random
HA: events A and B are correlated with each other.

Under H0, the probability P(A and B) = P(A)*P(B) = (1/365)^2
That is so unlikely that you can reject the null hypothesis with a high degree of confidence.

I'm sorry I can't recommend a text to use at the level you want. You might look for something that includes probability, the normal distribution, the sampling theorem, and hypothesis testing. Binomial distribution would be good too - but avoid anything fancier that these topics! [t-test, F-test, . . .]

Got it!

DrPhil said:

ok - this is another branch of statistics called "hypothesis testing." You set up what is called a "null hypothesis" and an "alternate hypothesis." The logic goers like this:
H0: events A and B are random
HA: events A and B are correlated with each other.

Under H0, the probability P(A and B) = P(A)*P(B) = (1/365)^2
That is so unlikely that you can reject the null hypothesis with a high degree of confidence.

I'm sorry I can't recommend a text to use at the level you want. You might look for something that includes probability, the normal distribution, the sampling theorem, and hypothesis testing. Binomial distribution would be good too - but avoid anything fancier that these topics! [t-test, F-test, . . .]

Click to expand...

Thanks. This is great. I just researched the hypothesis testing as you requested here and this is an excellent exercise for the class. This is what I am looking for.

prob2013 said:

Thanks. I think I need to go back and simplify the scenario. But, I think it is a scenario based on the probability of coincidences or rare events. I am trying to show that the coincidence is not a coincidence or random but specific and the couple experienced a intentional nuisance.

One way I guess would be to consider the events as independent. Any thoughts?

Click to expand...

Let's say the probability of being hacked at random on any given day is 0.1%. (It's obviously a lot lower than that in reality.)

The probability of being hacked exactly twice in a year by chance would be

\(\displaystyle \dbinom{365}{2} * 0.999^{363} * 0.001^2 \approx 4.61994\%.\)

In other words, being hacked at random exactly twice in a year would be fairly common: more than one person in twenty would be hacked AT LEAST twice each year.

The probability of being hacked at random on two specific days of the year would obviously be much smaller at \(\displaystyle 0.001^2 * 0.99^{363} \approx 0.00006\%.\)

But in this case, it is known that Gary was hacked twice.

The probabilty that he would be hacked at random on those two specific days given that he was hacked twice would be

\(\displaystyle \dfrac{0.001^2 * 0.99^{363} }{\binom{365}{2} * 0.999^{363} * 0.001^2} = \dfrac{2}{365 * 364}\approx 0.15053\% << 1\%.\)

It is virtually certain that Gary was the victim of an intentional tort.

Love it!

JeffM said:

Let's say the probability of being hacked at random on any given day is 0.1%. (It's obviously a lot lower than that in reality.)

The probability of being hacked exactly twice in a year by chance would be

\(\displaystyle \dbinom{365}{2} * 0.999^{363} * 0.001^2 \approx 4.61994\%.\)

In other words, being hacked at random exactly twice in a year would be fairly common: more than one person in twenty would be hacked AT LEAST twice each year.

The probability of being hacked at random on two specific days of the year would obviously be much smaller at \(\displaystyle 0.001^2 * 0.99^{363} \approx 0.00006\%.\)

But in this case, it is known that Gary was hacked twice.

The probabilty that he would be hacked at random on those two specific days given that he was hacked twice would be

\(\displaystyle \dfrac{0.001^2 * 0.99^{363} }{\binom{365}{2} * 0.999^{363} * 0.001^2} = \dfrac{2}{365 * 364}\approx 0.15053\% << 1\%.\)

It is virtually certain that Gary was the victim of an intentional tort.

Click to expand...

Thanks. This is another way at looking at it. I was looking at the theory of statistical significance and I believe this is it.