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📗 [3 points] There are two biased coins in my pocket: coin A has \(\mathbb{P}\left\{H | A\right\}\) = , coin B has \(\mathbb{P}\left\{H | B\right\}\) = . I took out a coin from the pocket at random with probability of A is . I flipped it three times (independently) and the outcome is . What is the probability that the coin was ?
📗 Answer: .
📗 [3 points] There are two biased coins in my pocket: coin A has \(\mathbb{P}\left\{H | A\right\}\) = , coin B has \(\mathbb{P}\left\{H | B\right\}\) = . I took out a coin from the pocket at random with probability of A is . I flipped it twice the outcome is . What is the probability that the coin was ?
📗 Answer: .
📗 [3 points] There are two biased coins in my pocket: coin A has \(\mathbb{P}\left\{H | A\right\}\) = , coin B has \(\mathbb{P}\left\{H | B\right\}\) = . I took out a coin from the pocket at random with probability of A is . I flipped it twice the outcome is . What is the probability that the coin was ?
📗 Answer: .
📗 [2 points] \(B\) is the boolean whether you have the bird flu or not. \(H\) is the boolean whether you have a headache or not. Let \(\mathbb{P}\left\{H = 1\right\}\) = , \(\mathbb{P}\left\{B = 1\right\}\) = , \(\mathbb{P}\left\{H = 0 | B = 1\right\}\) = . Given that you have the bird flu, what is the probability that you have headache?
📗 Answer: .
📗 [2 points] \(C\) is the boolean whether you have COVID-19 or not. \(F\) is the boolean whether you have a fever or not. Let \(\mathbb{P}\left\{F = 1\right\}\) = , \(\mathbb{P}\left\{C = 1\right\}\) = , \(\mathbb{P}\left\{F = 0 | C = 1\right\}\) = . Given that you have COVID-19, what is the probability that you have fever? Note: this question uses random fake data, please refer to CDC for actual data.
📗 Answer: .
📗 [3 points] Assume the prior probability of having a female child (girl) is the same as having a male child (boy) and both are 0.5. The Smith family has kids. One day you saw one of the Smith children, and she is a girl. The Wood family has kids, too, and you heard that at least one of them is a girl. What is the chance that the Smith family has a boy? What is the chance that the Wood family has a boy?
📗 Answer (comma separated vector): .
📗 [3 points] A tweet is ratioed if at least one reply gets more likes than the tweet. Suppose a tweet has replies, and each one of these replies gets more likes than the tweet with probability if the tweet is bad, and probability if the tweet is good. Given a tweet is ratioed, what is the probability that it is a bad tweet? The prior probability of a bad tweet is .
📗 Answer: .
📗 [3 points] You roll a 6-sided die times and observe the following counts in the table. Use Laplace smoothing (i.e. add-1 smoothing), estimate the probability of each side. Enter 6 numbers between 0 and 1, comma separated.
Side
1
2
3
4
5
6
Count
📗 Answer (comma separated vector): .
📗 [2 points] A traffic light repeats the following cycle: green seconds, yellow seconds, red seconds. A driver saw at a random moment. What is the probability that one second later the light became ?
📗 Answer: .
📗 [4 points] John tells his professor that he forgot to submit his homework assignment. From experience, the professor knows that students who finish their homework on time forget to turn it in with probability . She also knows that of the students who have not finished their homework will tell her they forgot to turn it in. She thinks that of the students in this class completed their homework on time. What is the probability that John is telling the truth (i.e. he finished it given that he forgot to submit it)?
📗 Answer: .
📗 [2 points] You have a coin that lands heads with probability . Flipping it times and they all happen to be heads. What is the probability that the next flips will contain one or more tails?
📗 Answer: .
📗 [4 points] Some Na'vi's don't wear underwear, but they are too embarrassed to admit that. A surveyor wants to estimate that fraction and comes up with the following less-embarrassing scheme: Upon being asked "do you wear your underwear", a Na'vi would flip a fair coin outside the sight of the surveyor. If the coin ends up head, the Na'vi agrees to say "Yes"; otherwise the Na'vi agrees to answer the question truthfully. On a very large population, the surveyor hears the answer "Yes" for fraction of the population. What is the estimated fraction of Na'vi's that don't wear underwear? Enter a fraction like 0.01 instead of a percentage 1%.
📗 Answer: .
📗 [4 points] Fill in the missing values in the following joint probability table so that A and B are independent.
-
A = 0
A = 1
B = 0
B = 1
??
??
📗 Answer (comma separated vector): .
📗 [2 points] In your day vacation, the counts of days are:
rainy
warm
bighorn (saw sheep)
days
N
N
N
N
N
Y
N
Y
N
N
Y
Y
Y
N
N
Y
N
Y
Y
Y
N
Y
Y
Y
Use maximum likelihood estimate (no smoothing), estimate the probability that P(bighorn = | rainy = , warm = )?
📗 Answer: .
📗 [4 points] Given the counts, find the maximum likelihood estimate of \(\mathbb{P}\left\{A = 1|B + C = s\right\}\), for \(s\) = .
A
B
C
counts
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
📗 Answer: .
📗 [3 points] Given two Boolean random variables, \(A\) and \(B\), where \(\mathbb{P}\left\{A\right\}\) = , \(\mathbb{P}\left\{B\right\}\) = , and \(\mathbb{P}\left\{A| \neg B\right\}\) = , what is \(\mathbb{P}\left\{A|B\right\}\)?
📗 Answer: .
📗 [3 points] Which of the following values of \(\mathbb{P}\left\{B\right\}\) is possible if \(\mathbb{P}\left\{A\right\} = \mathbb{P}\left\{A, B\right\}\) = ?
📗 Choices:
None of the above
📗 Calculator: .
📗 [2 points] We have a biased coin with probability of producing Heads. We create a predictor as follows: generate a random number uniformly distributed in (0, 1). If the random number is less than we predict Heads, otherwise, we predict Tails. What is this predictor's (expected) accuracy in predicting the coin's outcome?
📗 Answer: .
📗 [4 points] If \(\mathbb{P}\left\{A | B\right\}\) is times the value of \(\mathbb{P}\left\{B | A\right\}\), and \(\mathbb{P}\left\{A\right\}\) = . What is \(\mathbb{P}\left\{B\right\}\)?
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