optiver 2023 summer quant intern
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一开始的速算题目很简单这里不给大家罗列。
最难的概率题需要和大家分享一下
1. Consider a simple coin game where you can bet any amount of money. If it lands on Heads you double your bet; if it lands on Tails you lose your bet. Consider a strategy where you start by betting $\$ 1$. If you lose, you will bet $\$ 2$. If you lose again, you will bet $\$ 4$. If you lose again, you will bet $\$ 8$, and so on in this way until you finally win a toss. Once you have won a toss, your strategy has been implemented, and you stop. If you have a total bankroll of $\$ 63$ available to implement this strategy, what is your expected profit?
2. Two people start at opposite corners of a grid. Person A starts on the bottom left node. Person B starts on the top right node. Every second, Person A will randomly move either one edge to the right, or one edge up. At the same time, Person B will randomly move either one edge to the left, or one edge down. Each person must always move each second, for example, if Person A cannot move right anymore, they must move up. What is the probability that A and B share the same position at some point? Select the answer that is closest.
3. There are two chess players, Anish and Billy. You have a $20 \%$ chance of beating Anish in a match and an $80 \%$ chance of beating Billy in a match. You will play three games and if you wifn any two in a row, then you will win a prize. What is your chance of winning a prize if you play Anish, then Billy, then Anish? Select the answer that is closest.
4. On average, how many times must a dice be rolled until all sides have appeared at least once?
5. You roll five dice and sum their values. What is the probability that the sum is an even number?
6. You keep flipping a coin and whilst you do this you keep track of the number of Heads and the number of Tails. You can think of this as a race between Heads and Tails. For example, if after 9 flips you have landed on Heads 7 times and Tails 2 times, then Heads is winning the race by 5. Now consider a race of 3 flips. What is the probability that the side you flip first will be winning the race?
7. You roll three dice. What is the probability that two of these dice sum to an even number?
8. You throw one dice two times. What is the probability that the 2 nd throw has the same face value as the first throw?
另外上传的两个图，希望能帮助到大家！！！
找工作实习加油！！！💪
一开始的速算题目很简单这里不给大家罗列。
最难的概率题需要和大家分享一下
1. Consider a simple coin game where you can bet any amount of money. If it lands on Heads you double your bet; if it lands on Tails you lose your bet. Consider a strategy where you start by betting $\$ 1$. If you lose, you will bet $\$ 2$. If you lose again, you will bet $\$ 4$. If you lose again, you will bet $\$ 8$, and so on in this way until you finally win a toss. Once you have won a toss, your strategy has been implemented, and you stop. If you have a total bankroll of $\$ 63$ available to implement this strategy, what is your expected profit?
2. Two people start at opposite corners of a grid. Person A starts on the bottom left node. Person B starts on the top right node. Every second, Person A will randomly move either one edge to the right, or one edge up. At the same time, Person B will randomly move either one edge to the left, or one edge down. Each person must always move each second, for example, if Person A cannot move right anymore, they must move up. What is the probability that A and B share the same position at some point? Select the answer that is closest.
3. There are two chess players, Anish and Billy. You have a $20 \%$ chance of beating Anish in a match and an $80 \%$ chance of beating Billy in a match. You will play three games and if you wifn any two in a row, then you will win a prize. What is your chance of winning a prize if you play Anish, then Billy, then Anish? Select the answer that is closest.
4. On average, how many times must a dice be rolled until all sides have appeared at least once?
5. You roll five dice and sum their values. What is the probability that the sum is an even number?
6. You keep flipping a coin and whilst you do this you keep track of the number of Heads and the number of Tails. You can think of this as a race between Heads and Tails. For example, if after 9 flips you have landed on Heads 7 times and Tails 2 times, then Heads is winning the race by 5. Now consider a race of 3 flips. What is the probability that the side you flip first will be winning the race?
7. You roll three dice. What is the probability that two of these dice sum to an even number?
8. You throw one dice two times. What is the probability that the 2 nd throw has the same face value as the first throw?
另外上传的两个图，希望能帮助到大家！！！
找工作实习加油！！！💪
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