Statistics and Probability Answers

Dear sir/madam
I wonder if you can help... I must first say how difficult it has been to find a website for someone to hopefully answer my PROBABILITY question, but somehow iv'e been drawn to you !
For nearly a year iv'e been going to a local pub quiz[wales, u.k.] ..after the quiz a raffle is drawn and the winner of the ticket [which is around £55] has to guess a playing card [QUEEN OF HEARTS]to win a £1,000. Here's how it works..on the bar wall in the pub there's a glass cabinet just big enough to contain 52 playing cards - all the cards have their backs showing and secured with a drawing pin[i think that's a thumbtack in the States] and its all under lock and key. in fact, last week i won the raffle myself but guessed the wrong card-so no £1,000 !

But recently i became aware of this card raffle as a very intriguing PROBABILITY puzzle, so along with my friend who's in my team, we did 200 simulations between us and the result was fairly even..QH came up EVERYWHERE as one would expect. However, iv'e become aware of what seems like a gaping BIG PARADOX which I can't get to grips with..here it is: from the onset the probability of picking QH[in this case] is 51-1 against. The next go it's 50-1 against,then 49-1, etc. etc. Now, when you get halfway through the pack to the 26th card the probability is still a whopping 25-1 against you picking the QH... do you agree? But the PARADOX is this..if you have turned over 26 cards which is half of the 52 cards, then YOU MUST HAVE AN EQUAL CHANCE OF THE QH TO BE IN YOUR 26 CARD PILE!
This is what I dont get..everything is stacked against you from the first card at 51-1 against right up to the 26th card which is 25-1 against ........ yet, when you get to that stage everything tells you thet you have a 50-50 chance of the QH being in your first 26 card choice !!! How can something go from 25-1 to 50-50 in an instant ?
I know there's a discrepency somewhere but for the life of me I just cannot see it...in simple terms[as i'm no expert on probabilities] could you help me to understand what's going on here.
Thankyou
Yours Sincerely
Wyn

5. a) Discuss what is meant by Quality control and quality improvement.
b) What are the limitations of a quality control charts?

Suppose a sample of n = 25 items is drawn from a population described
by a Normal distribution with standard deviation σ = 15, and that the
sample mean of the 25 values is 96.4. Derive a 95% confidence interval
for μ, the (unknown) mean of this distribution, and use your confidence
interval to test, at the 5% significance level, the hypothesis H0 : μ = 100
against the general (two-sided) alternative H1 : μ
= 100.

1.A lawyer researched the average number of years served by 45 different justices on the Supreme Court. The average number of years served was 13.8 years with a standard deviation of 7.3 years. What is the 95% confidence interval estimate for the average number of years served by all Supreme Court justices? Place your limits, rounded to 1 decimal place, in the blanks. When entering your answer into the Final Exam Shell do not use any labels or symbols. Simply provide the numerical values. For example, 12.3 would be a legitimate entry.



Lower limit (first blank) = _ 11.7 _________, Upper limit (second blank) = ____15.9_______.

The probability of a cooker having a fault in the first year is 0.12.
a) what is the probability of a cooker not having in the first year? .
b) if the 2000 cookers are produced , how many would you expect to devolop a fault in the first year?

The probability that ea new battery does not work is 0.04.
a) what is the probability that e new battery does work?
b)A firm prodused 5000 batteries. How many would you expect to be faulty?

Two tigers are born at the zoo each year for 2 years.
a tiger is male or female

a) list the possible outcomes for the births after 2 years

The output voltage of power supplies manufactured by Belkin Products is believed to follow a normal distribution. Of primary concern to the firm is the average output voltage of a particular power supply unit, believed to be 10 volts. Eighteen observations taken at random from this unit are as given below:

10.85, 11.40, 10.81, 10.24, 10.23, 9.49, 9.89, 10.11, 10.57, 11.21, 10.10, 11.22, 10.31, 11.24, 9.51, 10.52, 9.92, 8.33

What is the 95% confidence interval for the average output voltage for this power supply unit?

& Suppose we draw a sample of size 150 from a lot of cash registers. We test each of the sampled cash registers and find 13 defectives.
a) Determine a 95% confidence interval for the proportion of defectives in the entire batch, and
b) Determine the sample size necessary to estimate the proportion of defectives to within 2% with 95% confidence.

20% of the deer in a wildlife preservation part are believed to have certain antler infections. A zoologist randomly captures 15 deer and checks their antlers. Assuming the prediction is accurate, determine the probability that more than three of the deer will have infected antlers.