The probability of an event or element in a sample space is written as $P(A)$, where $A$ is the event or element in the sample space. Answer the following problems symbolically and simplified. Each problem is worth 1 point.
[2 points] Our sample space contains three elements: $Q = \{1, 2, 3\}$. What is the probability of observing $P(1 \,\textrm{OR} \,2\, \textrm{OR}\, 3)$?
[2 points] What is the key differnce between $B_1$ and $B_2$ in these two expressions: $P( A \,\textrm{OR} \,B_1)$ and $P(A \,\textrm{AND}\, B_2)$ aside from the fact that OR is in the first expression and AND in the second.
[2 points] Consider the sample space of days in 2020 and assume each day has equal probability. If event $E$ is elements $\{\textrm{September 3rd}, \textrm{October 24th}, \textrm{December 13th}\}$, then what is the probability of the event $E$, $P(E)$?
[2 points] What is the probability of not event $E$ from the previous question?
[2 points] What is the probability of observing September 3rd and then September 4th?
[4 points] State the number of observations/samples for the probabilities specified in questions 1.3-1.5.
$B_1$ is another possible element from the same observation as $A$ in the OR expression. $B_2$ is for a second sequential independent sample. Another possible answer is stating that A and $B_1$ must be in the same sample space, where $B_2$ could be a different sample space than $A$.
You are rolling a die with 6 sides. Event $A$ is that you roll a 4. Event $B$ is that you roll a $6$. Event $C$ is rolling a number greater than 2. Answer these problems using python code. Each problem is worth 1 point.
The markdown answers are only for clarity and not required for credit
$$ A = \{4\} $$1 / 6
0.16666666666666666
The first roll has no impact
$$ P(A\,\textrm{OR}\, C) = P(A) + P(C) - P(A \cap C) $$1 / 6 + 4 / 6 - 1/6
0.6666666666666666
The wording means we could roll A then C or C then A. A combination.
(1 / 6) * (4 / 6) + (4 / 6) * (1 / 6)
0.2222222222222222
Again, the roll has no impact
1/6
0.16666666666666666
The wording means only one permutation: A then C
(1 / 6) * ( 1 / 6)
0.027777777777777776
2.6
We can compute $P(\sim A)$ and compute probability of that occuring 10 times in a row and negate that. Put another way, we know NOT (NOT A AND NOT A AND NOT A AND NOT A AND NOT A...) is the same thing as A occuring at least once.
1 - (5 / 6)**10
0.8384944171101543
Answers these problems symbolically
Each problem is worth 4 points
Can use anything reasonable instead of $\infty$ $$ \{0, 1, 2, \ldots, \infty\} $$
W - working, B - broken. It's ok if they assume markers are distinguishable, but they should have a much bigger sample space, $2^3 + 2^2 + 2^1 + 2^0 = 15$
$$ \{ (), (W), (B), (WW), (WB), (BB), (WWW), (WWB), (WBB), (BBB)\} $$The length of the password raises the power which increases the number of possible passwords much more than changing the base of the exponent. So it is better to have longer passwords than silly rules.
Choose two random things to observe over the course of the semester. Record your data youself and you should end up with at least 10 observations per month. Examples could be: screen time per day, time it takes to get to class, number of people you eat lunch with, colors in your outfit, number of birds in a particular tree, temperature at 9am.
[8points] Write out the sample space of your two data and give an example of them.