A restaurant offers apple and blueberry pies and stocks an equal number of each kind of pie. Each day ten customers request pie. They choose, with equal probabilities, one of the two kinds of pie. How many pieces of each kind of pie should the owner provide so that the probability is about .95 that each customer gets the pie of his or her own choice?

A poker hand is a set of 5 cards randomly chosen from a deck of 52 cards. Find the probability of a

• royal flush (ten, jack, queen, king, ace in a single suit).
• straight flush (five in a sequence in a single suit, but not a royal flush).
• four of a kind (four cards of the same face value).
• full house (one pair and one triple, each of the same face value).
• flush (five cards in a single suit but not a straight or royal flush).
• straight (five cards in a sequence, not all the same suit). (Note that in straights, an ace counts high or low.)

If a set has [math]2n[/math] elements, show that it has more subsets with [math]n[/math] elements than with any other number of elements.

Let [math]b(2n,.5,n)[/math] be the probability that in [math]2n[/math] tosses of a fair coin exactly [math]n[/math] heads turn up. Using Stirling's formula (Theorem), show that [math]b(2n,.5,n)\sim 1/\sqrt{\pi n}[/math]. Use the program BinomialProbabilities to compare this with the exact value for [math]n = 10[/math] to 25.

A baseball player, Smith, has a batting average of [math].300[/math] and in a typical game comes to bat three times. Assume that Smith's hits in a game can be considered to be a Bernoulli trials process with probability .3 for success. Find the probability that Smith gets 0, 1, 2, and 3 hits.

The Siwash University football team plays eight games in a season, winning three, losing three, and ending two in a tie. Show that the number of ways that this can happen is

Using the technique of Exercise, show that the number of ways that one can put [math]n[/math] different objects into three boxes with [math]a[/math] in the first, [math]b[/math] in the second, and [math]c[/math] in the third is [math]n!/(a!\,b!\,c!)[/math].

Baumgartner, Prosser, and Crowell are grading a calculus exam. There is a true-false question with ten parts. Baumgartner notices that one student has only two out of the ten correct and remarks, “The student was not even bright enough to have flipped a coin to determine his answers.” “Not so clear,” says Prosser. “With 340 students I bet that if they all flipped coins to determine their answers there would be at least one exam with two or fewer answers correct.” Crowell says, “I'm with Prosser. In fact, I bet that we should expect at least one exam in which no answer is correct if everyone is just guessing.” Who is right in all of this?

A gin hand consists of 10 cards from a deck of 52 cards. Find the probability that a gin hand has

• all 10 cards of the same suit.
• exactly 4 cards in one suit and 3 in two other suits.
• a 4, 3, 2, 1, distribution of suits.

A six-card hand is dealt from an ordinary deck of cards. Find the probability that:

• All six cards are hearts.
• There are three aces, two kings, and one queen.
• There are three cards of one suit and three of another suit.