Question

You roll a fair 6-sided die two times and get paid the higher of the two rolls in dollars if the rolls are different. If they are the same, you get paid $0. Compute your expected payoff from this game.

Input/Output Example

Total outcomes = 36  
Let (i, j) be the result of two rolls

If i == j: payoff = $0  
If i ≠ j: payoff = max(i, j)

Sum over all outcomes, compute expected value:
E = (1/36) * [Sum over i ≠ j of max(i, j)]

After full computation:
E ≈ 4.47

Question

Assume that in a bakery, each customer buys only one item at a time.
There is a 70% chance a customer will buy a croissant and a 30% chance a customer will buy a muffin.
There are only 2 muffins left and 5 people are still waiting in line.
Compute the probability that these two muffins will be sufficient, i.e., no customer will want a muffin and find that there are none left.

Input/Output Example

Let X be the number of customers who want muffins ~ Binomial(n=5, p=0.3)

We need P(X ≤ 2)

P(X = 0) = C(5,0)*(0.3)^0*(0.7)^5 = 0.16807  
P(X = 1) = C(5,1)*(0.3)^1*(0.7)^4 = 5*0.3*0.2401 = 0.36015  
P(X = 2) = C(5,2)*(0.3)^2*(0.7)^3 = 10*0.09*0.343 = 0.3087  

P(X ≤ 2) = 0.16807 + 0.36015 + 0.3087 = 0.83692

Answer: 84%

Question

Asta, Bronya, and Clara all have different numbers of cats. They had the following conversation.

i. Bronya to Clara: You have the most cats.
ii. Asta to Bronya: I have exactly 30% more cats than you.
iii. Asta to Clara: The number of cats you have is the average between the number of cats I have and the number of cats Bronya has.
iv. Clara to Asta: You have at least 4 more cats than me.

However, not all of these statements are true.
When speaking to someone with fewer cats, the speaker always tells the truth.
When speaking to someone with more cats, the speaker always lies.
How many cats does Asta have?

Input/Output Example

Try values for (Asta, Bronya, Clara), test all 4 statements and the truth/lie logic  
Check who lies/truths based on cat count comparison  

Through systematic testing and consistent logic:
Asta has 13 cats

Question

Two friends canoe upstream for 4 hours, only to realize that their campsite is downstream.
They turn around and paddle downstream for 5 hours.
The next day, they pack up and canoe back to their original starting point 23 miles upstream and arrive at 16:00.
Assume that the river always flows at a constant rate of 2 miles per hour and the two friends always paddle at a constant rate.
What time did they leave for their return trip?

Input/Output Example

Let p = paddling speed in still water  
Upstream speed = p - 2  
Distance = 23 miles

Time = Distance / (p - 2) = T  
Need to solve T such that 16:00 - T = departure time

From day 1:  
Upstream 4 hr + Downstream 5 hr = 9 hr  
Let distance d:  
4(p - 2) = d  
5(p + 2) = d  
=> 4(p - 2) = 5(p + 2) ⇒ p = 18

Return trip: upstream 23 miles at speed 16 mph ⇒ time = 23 / (18 - 2) = 23 / 16 = 1.4375 hr = 1 hr 26 min

16:00 - 1:26 = 14:34  
Answer: 14:34

Question

You baked 5 indistinguishable snickerdoodle cookies and 6 indistinguishable chocolate chip cookies.
Compute the number of ways to arrange 6 of these cookies into a straight line.

Input/Output Example

Choose 6 cookies out of 11 (5 S, 6 C) with repetition allowed within type, but distinguishable by type

Possible counts:
(0S,6C): 1  
(1S,5C): 6  
(2S,4C): 15  
(3S,3C): 20  
(4S,2C): 15  
(5S,1C): 6

Total = sum of combinations:
C(6,0) + C(6,1) + C(6,2) + C(6,3) + C(6,4) + C(6,5) = 63

我们长期稳定承接各大科技公司如TikTok、Google、Amazon等的OA笔试代写服务，确保满分通过。如有需求，请随时联系我们。

We consistently provide professional online assessment services for major tech companies like TikTok, Google, and Amazon, guaranteeing perfect scores. Feel free to contact us if you're interested.