Probability Tes...

A biased coin (with probability of obtaining a Head equal to $$p > 0$$) is tossed repeatedly and independently until the first head is observed. Compute the probability that the first head appears at an even numbered toss.

A factory production line is manufacturing bolts using three machines, A, B and C. Of the total output, machine A is responsible for $$25$$%, machine B for $$35$$% and machine C for the rest. It is known from previous experience with the machines that $$5$$% of the output from machine A is defective, $$4$$% from machine B and $$2$$% from machine C. A bolt is chosen at random from the production line and found to be defective. What is the probability that it came from machine B 

From a batch of $$100$$ items of which $$20$$ are defective, exactly two items are chosen, one at a time, without replacement. Calculate the probabilities that both items chosen are defective

While watching a game of Champions League football in a cafe, you observe someone who is clearly supporting Manchester United in the game. What is the probability that they were actually born within 25 miles of Manchester? Assume that:

$$\implies $$the probability that a randomly selected person in a typical local bar environment is born within $$25$$ miles of Manchester is $$\dfrac{1}{20}$$, and;

$$\implies$$the chance that a person born within $$25$$ miles of Manchester actually supports United is $$\dfrac{7}{10}$$; 

$$\implies$$the probability that a person not born within $$25$$ miles of Manchester supports United with probability $$\dfrac{1}{10}$$

A garage mechanic keeps a box of good springs to use as replacements on customers cars. The box contains $$5$$ springs. A colleague, thinking that the springs are for scrap, tosses three faulty springs into the box. The mechanic picks two springs out of the box while servicing a car. Find the probability that the second spring drawn is faulty.

Tossing a coin is an example of .........

For a random experiment, all possible outcomes are called

Two players A and B are competing at a trivia quiz game involving a series of questions. On any individual question, the probabilities that A and B give the correct answer are $$\alpha$$ and $$\beta$$ respectively, for all questions, with outcomes for different questions being independent. The game finishes when a player wins by answering a question correctly. 

Compute the probability that A wins if A answers the first question

Two players A and B are competing at a trivia quiz game involving a series of questions. On any individual question, the probabilities that A and B give the correct answer are $$\alpha$$ and $$\beta$$ respectively, for all questions, with outcomes for different questions being independent. The game finishes when a player wins by answering a question correctly. 
Compute the probability that A wins if B answers the first question.

Sample space for experiment in which a dice is rolled is