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# Tag Archives: JCM_math230_HW4_F22

## Which deck is rigged ?

Two decks of cards are sitting on a table. One deck is a standard deck of 52 cards. The other deck (called the rigged deck) also has 52 cards but has had 4 of the 13 Harts replaced by Diamonds. (Recall that a standard deck has 4 suits: Diamonds, Harts, Spades, and Clubs. normal there are 13 of each suit.)

- What is the probability one chooses 4 cards from the rigged deck and gets exactly 2 diamonds and no hearts?
- What is the probability one chooses 4 cards from the standard deck and gets exactly 2 diamonds and no hearts?
- You randomly chose one of the decks and draw 4 cards. You obtain exactly 2 diamonds and no hearts.
- What is the probability you chose the cards from the rigged deck?
- What is the probability you chose the cards from the standard deck?
- If you had to guess which deck was used, which would you guess? The standard or the rigged ?

## Defective Machines

Suppose that the probability that an item produced by a certain machine will be defective is 0.12.

- Find the probability (exactly) that a sample of 10 items will contain at most 1 defective item.
- Use the Poisson to approximate the preceding probability. Compare your two answers.

[Inspired Ross, p. 151, example 7b ]

## Cards again

Given a well shuffled standard deck of 52 cards, what is the probability of what of the following events. (Think before you jump.)

- The 1st card is an ace.
- The 15th card is an ace.
- The 9th card is a diamond.
- The last 5 cards are hearts.
- The 17th card is the ace of diamonds and the 14 is the King of spades
- The 5th card is a diamond given that the 50th card is a diamond.

## Approximation: Rare vs Typical

Let \(S\) be the number of successes in 25 independent trials with probability \(\frac1{10}\) of success on each trial. Let \(m\) be the most likely value of S.

- find \(m\)
- find the probability that \(\mathbf{P}(S=m)\) correct to 3 decimal places.
- what is the normal approximation to \(\mathbf{P}(S=m)\) ?
- what is the Poisson approximation to \(\mathbf{P}(S=m)\) ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25. Would the normal or Poisson approximation give a better approximation in this case ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25 and the probability of success as \(\frac1{1000}\) rather that \(\frac1{10}\) . Would the normal or Poisson approximation give a better approximation in this case ?

[Pitman p122 # 7]

## Poker Hands: counting

Assume that each of Poker hands are equally likely. The total number of hands is

\[\begin{pmatrix} 52 \\5\end{pmatrix}\]

Find the probability of being dealt each of the following:

- a straight flush ( all cards of the same suit and in order)
- a regular straight (but
**not**a flush) - two of a kind
- four of a kind
- two pairs (but
**not**four of a kind) - a full house (a pair and three of a kind)

In all cases, we mean exactly the hand stated. For example, four of a kind does not count as 2 pairs and a full house does not count as a pair or three of a kind.

## The matching problem

There are \(n\) letters addressed to \(n\) eople at different addresses. The \(n\) addresses are typed on \(n\) envelopes. A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.

- Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
- What is the probability approximately, for large \(n\) ?

[ For example, the needed inclusion-exclusion formula is given in Problem 12, p. 31 in Pitman]

You will also need to know the number of elements in the set

\[ \{ (i_1,i_2,\cdots, i_k) : 1 \leq i_1 < i_2 < \cdots < i_k\leq n\} \]

which is discussed here.