Q1
A fruit vendor has a basket containing 30 apples and 30 oranges. You play a game where you draw two fruits at random from the basket without replacement. The rules of the game are as follows:
- If both fruits are apples, you get 1 point.
- If both fruits are oranges, the vendor gets 1 point.
- If the fruits are different, no one gets a point.
The game ends when all fruits have been drawn. If you have more points than the vendor at the end of the game, you win $100. If you lose, you get nothing. If you tie, then the vendor forces you to play in a rematch game with the exact same setup.
In order to play the game in the first place, you must pay the vendor some amount of money. What is the maximum amount of money you should be willing to pay to play this game if you want to break even on average?
Q2
You decide to bet $100 on your favorite team in a best-of-7 baseball series, where the first team to win 4 games wins the series. The odds for each game are even, meaning each team has a 50% chance of winning any given game.
If your team wins the series, you are returned your $100 bet and rewarded an additional $200. If they lose, you lose your $100 bet.
What is the expected value of your bet at the start of the series?
Q3
You roll a fair eight-sided die. If the die shows an odd number, you then roll a fair six-sided die.
What is the probability that the sum of the numbers rolled is either 7 or 8?
Q4
There are seven nickels and four dimes in your pocket. Five of the nickels and two of the dimes are Canadian. The others are US currency. You randomly select two coins from your pocket without replacement.
What is the probability that at least one of the selected coins is a nickel or Canadian currency?
Q5
There are three types of machines in a factory: Type A, Type B, and Type C. Each machine contributes differently to the total production output of the factory.
The factory's production target is met if and only if the number of Type A machines plus twice the number of Type B machines plus three times the number of Type C machines equals 12.
Assume that the number of Type A, B, and C machines is independently chosen with equal probability from the set {0, 1, 2, 3, 4}. In machine learning, a problem is said to be linearly separable if there exists a linear boundary that can separate the data points into two classes without error.
Which of the following is true about the classification problem of meeting the production target as either possible or impossible?
- It is never linearly separable.
- It can be linearly separable if and only if we have at most one data point for each combination of the numbers of Type A, Type B, and Type C machines.
- It is always linearly separable.
- It can be linearly separable if and only if we have at least one data point for each combination of the numbers of Type A, Type B, and Type C machines.
- It can be linearly separable if and only if we have exactly one data point for each combination of the numbers of Type A, Type B, and Type C machines.
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