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Chapter 2: Q. 2.27 (page 50)
An urn contains red and black balls. Playerswithdraw balls from the urn consecutively until a red ball is selected. Find the probability that selects the red
ball. (draws the first ball, then, and so on. There is no replacement of the balls drawn.)
The probability that A selects the red ball is.
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How many people have to be in a room in order that the probability that at least two of them celebrate their birthday in the same month is at least Assume that all possible monthly outcomes are equally likely.
Use Venn diagrams
to simplify the expressions ;
to prove DeMorgan鈥檚 laws for eventsand. [That is, prove, and
Consider the following technique for shuffling a deck of n cards: For any initial ordering of the cards, go through the deck one card at a time, and at each card, flip a fair coin. If the coin comes up heads, then leave the card where it is; if the coin comes up tails, then move that card to the end of the deck. After the coin has been flipped n times, say that one round has been completed. For instance, if the initial ordering isthen if the successive flips result in the outcome then the ordering at the end of the round is Assuming that all possible outcomes of the sequence of coin flips are equally likely, what is the probability that the ordering after one round is the same as the initial ordering?
An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The
classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students who are in both Spanish and French, 4 who are in both Spanish and German, and 6 who are in both French and German. In addition, there are 2 students taking all 3 classes.
(a) If a student is chosen randomly, what is the probability that he or she is not in any of the language classes?
(b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?
(c) If 2 students are chosen randomly, what is the probability that at least 1 is taking a language class?
If and, show that.In general, prove Bonferroni鈥檚 inequality, namely.
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