1. Suppose that an urn contains 8 red balls and 4 white balls. We draw 2 balls from the urn without replacement, (a) If we assume that at each draw, each ball in the urn is equally likely to be chosen, what is the probability that both balls drawn are red? (b) Now suppose that the balls have different weights, with each red ball having weight r and each white ball having weight w. Suppose that the probability that a given ball in the urn is the next one selected is its weight divided by the sum of the weights of all balls currently in the urn. Now what is the probability that both balls are red?
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
1. Suppose that an urn contains 8 red balls and 4 white balls. We draw 2 balls from the
urn without replacement, (a) If we assume that at each draw, each ball in the urn is
equally likely to be chosen, what is the probability that both balls drawn are red? (b)
Now suppose that the balls have different weights, with each red ball having weight r
and each white ball having weight w. Suppose that the probability that a given ball in
the urn is the next one selected is its weight divided by the sum of the weights of all
balls currently in the urn. Now what is the probability that both balls are red?
2. Consider the following game played with an ordinary deck of 52 playing cards: The
cards are shuffled and then turned over one at a time. At any time, the player can
guess that the next card to be turned over will be the ace of spades; if it is, then the
player wins. In addition, the player is said to win if the ace of spades has not yet
appeared when only one card remains and no guess has yet been made. What is a good
strategy? What is a bad strategy?
3. A contestant on a quiz show is presented with two questions, questions 1 and 2, which
he is to attempt to answer in some order he chooses. If he decides to try question i first,
then he will be allowed to go on to question j,j ̸= i, only if his answer to question i is
correct. If his initial answer is incorrect, he is not allowed to answer the other question.
The contestant is to receive Vi dollars if he answers question i correctly, i = 1,2. For
instance, he will receive V1 + V2 dollars if he answers both questions correctly. If the
probability that he knows the answer to question i is Pi
, i = 1,2, which question should
he attempt to answer first so as to maximize his expected winnings? Assume that the
events Et,i = 1,2, that he knows the answer to question i are independent events.
4. Consider a jury trial in which it takes 8 of the 12 jurors to convict the defendant; that is,
in order for the defendant to be convicted, at least 8 of the jurors must vote him guilty.
If we assume that jurors act independently and that whether or not the defendant is
guilty, each makes the right decision with probability θ, what is the probability that
the jury renders a correct decision?
5. Ten percent of the tools produced in a certain manufacturing process turn out to be
defective. Find the probability that in a sample of 10 tools chosen at random exactly two
will be defective, by using (a) the binomial distribution, (b) the Poisson approximation
to the binomial distribution.
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