Note: Answer 2 out of the 3 problems. Solutions are due Sunday 20th at 9 pm.
Note 2: You are only allowed to use code that is developed in class. For problem 1, you are not allowed to use numpy's integration features.
Only using numpy and matplotlib, write a self contained program that can find the area trapped between the following two curves.
and
$$ g(x) = b_1x^3 + b_2x^2+b_3x+b_4 $$for any $ a_1,a_2,a_3, b_1,b_2,b_3,b_4 \in \mathbb{R} $
The prime 41, can be written as the sum of six consecutive primes:
$$ 41 = 2 + 3 + 5 + 7 + 11 + 13 $$This is the longest sum of consecutive primes that adds to a prime below one-hundred.
Which prime, below one-million, can be written as the sum of the most consecutive primes?
Julie proposes the following wager to her sister Louise. She suggests they play a game of chance to determine who will wash the dishes. For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1. The game starts with S = 0. The first player, Louise, adds to S different random numbers from the generator until S > 1 and records her last random number 'x'. The second player, Julie, continues adding to S different random numbers from the generator until S > 2 and records her last random number 'y'. The player with the highest number wins and the loser washes the dishes, i.e. if y > x the second player wins.
For example, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 1 and x = 0.44. If the second players draws 0.1, 0.27 and 0.91, the second player turn ends since 0.62+0.44+0.1+0.27+0.91 > 2 and y = 0.91. Since y > x, the second player wins.
Louise thinks about it for a second, and objects: "That's not fair". What is the probability that the second player wins? Give your answer rounded to 10 places.
Remember that $P(E) = \frac{N(E)}{N(S)}$
that is,
Probability of something happening is the number of times it happened divided by the number in the sample space.