Probability Examples and Explanations

A gambling book recommends the following “winning strategy” for the game of roulette:
Bet $1 on red. If red appears (which has probability $\frac{18}{38}$,then take the $1 profit and quit.
If red does not appear and you lose this bet ( which has probability $\frac{20}{38}$ of occurring), make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit.
Let X denote your winnings when you quit.
(a) Find P{X>0}.
(b) Are you convinced that the strategy is indeed a “winning” strategy?
(c) Find E[X].

Express the confidence interval using the indicated format. (Based on the percentages of red, orange, yellow, and blue MMs, the confidence intervals are calculated.) 

Express 0.0434<p<0.217 in the form of$\hat{p}\pm E$ 

When system reliability is increased through redundant or backup components, the redundancy principle is applied.
Assume that your alarm clock has a 0.975 probability of working on any given morning. 
With one alarm clock, you have a 0.975 probability of being awakened. 
What is the probability of being awakened if you use two alarm clocks?

The probability density function of the net weight in pounds of a packaged chemical herbicide is $f(x)=2.0$ for $49.75<x<50.25$ pounds.
a. Determine the probability that a package weighs more than 50 pounds.
b. How much chemical is contained in 90% of all packages?

Find the following probabilities for the standard normal variable z:
(a) ${\displaystyle P(-1.96\le z\le 1.96)}$
(b) $P(z>1.96)$

(c) $P(z<-1.96)$

An urn contains 3 red and 7 black balls. Players A and B withdraw balls from the urn consecutively until a red ball is selected. Find the probability that A selects the red ball. (A draws the first ball, then B, and so on. There is no replacement of the balls drawn.)

A company with a fleet of 150 vehicles discovered that seven out of the 22 emissions systems they evaluated did not adhere to pollution control standards. Is this conclusive proof that up to 20% of the fleet may not be in compliance? Test a relevant theory, then provide your findings. Before moving on, confirm that all necessary presumptions and criteria have been met.

The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three. (1) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let S, W, and R be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let X denote the number of solutions that are possible after that player observes his or her three cards. (2) Express X in terms of S, W, and R. (3) Find E[X]