# Tag Archives: JCM_math230_HW4_S15

## Defective Machines

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

1.  Find the probability (exactly)  that a sample of 10 items will contain at most 1 defective item.
2. 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.)

1. The 1st card is an ace.
2. The 15th card is an ace.
3. The 9th card is a diamond.
4. The last 5 cards are hearts.
5. The 17th card is the ace of diamonds and the 14 is the King of spades
6. 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.

1. find $$m$$
2. find the probability that  $$\mathbf{P}(S=m)$$ correct to 3 decimal places.
3. what is the normal approximation to $$\mathbf{P}(S=m)$$  ?
4. what is the Poisson approximation to $$\mathbf{P}(S=m)$$ ?
5. 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 ?
6. 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:

1. a straight flush ( all cards of the same suit and in order)
2. a regular straight (but not a flush)
3. two of a kind
4. four of a kind
5. two pairs (but not four of a kind)
6. 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.

1. Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
2. 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.

## Leukemia Test

A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time
in patients younger than 55. A test group has 17 patients 55 and older and 12 patients younger than 55.

1. A uniformly random patient is chosen from the test group, and the drug is administered and it is a success. What is the probability the patient was 55 and older?
2. A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all of them?