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There were ten red bottles sitting on the wall. The probability of a red bottle accidentally falling is 0.95. What is the probability that fewer than 8 of the green bottles accidentally fall?


You consult Joe the bookie as to the form in the 2.30 at Ayr. He tells you that, of 16 runners, the favourite has probability 0.3 of winning, two other horses each have probability 0.20 of winning, and the remainder each have probability 0.05 of winning, excepting Desert Pansy, which has a worse than no chance of winning. What do you think of Joe’s advice?


M&M sweets are of varying colours and the different colours occur in different proportions. The table below gives the probability that a randomly chosen M&M has each colour, but the value for tan candies is missing.

Colour Brown Red Yellow Green Orange Tan

Probability 0.3 0.2 0.2 0.1 0.1 ?

(a) What value must the missing probability be?

(b) You draw an M&M at random from a packet. What is the probability of each of the following events?

i. You get a brown one or a red one.

ii. You don’t get a yellow one.

iii. You don’t get either an orange one or a tan one.

iv. You get one that is brown or red or yellow or green or orange or tan.


A bag contains fifteen balls distinguishable only by their colours; ten are blue and five are red. I reach into the bag with both hands and pull out two balls (one with each hand) and record their colours.

(a) What is the random phenomenon?

(b) What is the sample space?

(c) Express the event that the ball in my left hand is red as a subset of the sample space.


A fair coin is tossed, and a fair die is thrown. Write down sample spaces for

(a) the toss of the coin;

(b) the throw of the die;

(c) the combination of these experiments.

Let A be the event that a head is tossed, and B be the event that an odd number is thrown. Directly from the sample space, calculate P(A ∩ B) and P(A ∪ B).


Describe the sample space and all 16 events for a trial in which two coins are thrown and each shows either a head or a tail.

What are applications of calculus in economics

2.2. Solve the ivp sin(x) dx + y dy = 0, where y(0)

2.1. Solve 2xy + 6x + (x^2 - 4)y'=0

Let x(s) be a curve with arc length parametrization, and satisfies "||x(s)||\\leq ||x(s_0)||\\leq1" for all s sufficiently close to x0. Prove "\\kappa" (s0) > 1. (Hint: Consider f(s) = ||x(s)|| 2 . Then f(s) has a local maximum at s0. Calculate f''(s0))

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