Chapter 12

Four cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of exactly one pair?

In how many ways can two aces and three kings be selected from a standard deck of cards if cards are drawn without replacement?

An urn contains 12 green and 24 blue balls. (a) You take 10 balls out of the urn without replacing them. Find the probability that 6 of the 10 balls are blue. (b) You take a ball out of the urn, note its color, and replace it. You withdraw a total of 10 balls this way. Find the probability that 6 of the 10 balls are blue.

Let \(X\) be exponentially distributed with parameter \(\lambda\). Find \(E(X)\)

Thirteen cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability that all are red?

In the game of poker, determine the number of ways exactly two pairs can be picked.

Sampling With and Without Replacement An urn contains \(K\) green and \(N-K\) blue balls. (a) You take \(n\) balls out of the urn. Find the probability that \(k\) of the \(n\) balls are green. (b) You take a ball out of the urn, note its color, and replace it. You repeat these steps \(n\) times. Find the probability that \(k\) of the \(n\) balls are green.

Let \(X\) be exponentially distributed with parameter \(\lambda\). Find \(\operatorname{var}(X)\).

Four cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability that all are of different suits?

In the game of poker, determine the number of ways a flush (five cards of the same suit) can be picked.

Suppose that the lifetime of a battery is exponentially distributed with an average life span of three months. What is the probability that the battery will last for more than four months?

Five cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of exactly two pairs?

In the game of poker, determine the number of ways four of a kind (four cards of the same value, plus one other card) can be picked.

Suppose that the lifetime of a battery is exponentially distributed with an average life span of two months. You buy six batteries. What is the probability that none of them will last more than two months? (Assume that the batteries are independent.)

Five cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of three of a kind and a pair (for instance, Q Q Q 3 3)? (This is called a full house in poker.)

In the game of poker, determine the number of ways a straight (five cards with consecutive values, such as A 2345 or 7 \(8910 \mathrm{~J}\), but not necessarily all of the same suit) can be picked.

Suppose that the lifetime of a radioactive atom is exponentially distributed with an average life span of 27 days. (a) Find the probability that the atom will not decay during the first 20 days after you start to observe it. (b) Suppose that the atom does not decay during the first 20 days that you observe it. What is the probability that it will not decay during the next 20 days?

Counterpoint is a musical term that means the combination of simultaneous voices; it is synonymous with polyphony. In triple counterpoint, three voices are arranged such that any voice can take any place of the three possible positions: highest, intermediate, and lowest voice. In how many ways can the three voices be arranged?

If \(X\) has distribution function \(F(x)\), we can show that \(F(X)\) is uniformly distributed over the interval \((0,1)\). Use this fact to generate exponentially distributed random variables with mean 1. [Assume that a computer generated the following four uniformly distributed random variables on the interval \((0,1): 0.0371\), \(0.5123,0.1370,0.9865 .]\)

Counterpoint is a musical term that means the combination of simultaneous voices; it is synonymous with polyphony. In quintuple counterpoint, five voices are arranged such that any voice can take any place of the five possible positions: from highest to lowest voice. In how many ways can the five voices be arranged?

An urn contains six green, eight blue, and 10 red balls. You take one ball out of the urn, note its color, and replace it. You withdraw a total of six balls this way. What is the probability that you sampled two of each color?

Suppose the number of customers per hour arriving at the post office is a Poisson process with an average of four customers per hour. (a) Find the probability that no customer arrives between 2 and 3 ?.?. (b) Find the probability that exactly two customers arrive between 3 and 4 P.M. (c) Assuming that the number of customers arriving between 2 and 3 P.M. is independent of the number of customers arriving between 3 and 4 p.M., find the probability that exactly two customers arrive between 2 and 4 P.M. (d) Assume that the number of customers arriving between 2 and 3 P.m. is independent of the number of customers arriving between 3 and 4 p.m. Given that exactly two customers arrive between 2 and 4 P.M., what is the probability that both arrive between 3 and 4 P.M.?

An urn contains eight green, four blue, and six red balls. You take one ball out of the urn, note its color, and replace it. You repeat these steps four times. What is the probability that you sampled two green, one blue, and one red ball?

Suppose the number of customers per hour arriving at the post office is a Poisson process with an average of five customers per hour. (a) Find the probability that exactly one customer arrives between 2 and 3 P.M. (b) Find the probability that exactly two customers arrive between 3 and 4 P.M. (c) Assuming that the number of customers arriving between 2 and 3 P.M. is independent of the number of customers arriving between 3 and 4 p.m., find the probability that exactly three customers arrive between 2 and 4 P.M. (d) Assume that the number of customers arriving between 2 and 3 P.m. is independent of the number of customers arriving between 3 and 4 p.m. Given that exactly three customers arrive between 2 and 4 p.m., what is the probability that one arrives between 2 and 3 p.m. and two between 3 and 4 P.m.?

You arrive at a bus stop at a random time. Assuming that busses arrive according to a Poisson process with rate \(4 / \mathrm{hr}\), what is the expected time to the next arrival?

Assume that \(N(t)\) is a Poisson process with rate \(\lambda\) and \(T_{1}\) is the time of the first arrival. Show that, for \(s

A number of traits are caused by recessive genes. The traits show up only in individuals who are homozygous (i.e., have two copies of the mutant gene). An individual with one normal and one mutant gene is a carrier, but does not exhibit the trait. Calculate each of the probabilites. The inability to roll one's tongue is caused by a single pair of recessive genes \((r r) .\) For a couple consisting of a heterozygote individual \((R r)\) and an affected person \((r r)\), what is the probability that, among their four children, at most one child is unable to roll his or her tongue?

Suppose the lifetime of a lightbulb is exponentially distributed with mean 3 years. The lightbulb is instantly replaced upon failure. (a) Find the probability that the lightbulb will have failed after two years. (b) What is the probability that, over a period of five years, the lightbulb was replaced only once?

An attached earlobe is caused by a single pair of recessive genes \((a a) .\) For a couple consisting of a heterozygous individual ( \(A a\) ) and an affected person ( \(a a\) ), what is the probability that a child has an unattached earlobe?

Suppose the lifetime of a light bulb is exponentially distributed with mean 1 year. The light bulb is instantly replaced upon failure. What is the probability that, over a period of five years, at most five light bulbs are needed?

A number of traits are caused by recessive genes. The traits show up only in individuals who are homozygous (i.e., have two copies of the mutant gene). An individual with one normal and one mutant gene is a carrier, but does not exhibit the trait. Calculate each of the probabilites. Tay-Sachs disease is caused by a single pair of recessive genes. If both parents are carriers of the mutant gene, what is the likelihood that none of their four children will be affected?

Suppose the lifetime of a laptop computer is exponentially distributed with mean five years. (a) Find the probability that the computer will have failed after three years. (b) Given that the computer has worked for six years, find the probability that it will work for another year.

Assume a \(1: 1\) sex ratio. A woman who is a carrier of hemophilia has two daughters and two sons with a man who is not hemophilic. What is the probability that one daughter is not a carrier, one daughter is a carrier, one son is hemophilic, and one son is not hemophilic?

Suppose the lifetime of an organism is exponentially distributed with hazard rate function \(\lambda(x)=2 /\) day. (a) Find the probability that an individual of this species lives for more than three days. (b) What is the expected lifetime?

A random experiment consists of flipping a fair coin until the first time heads appears. Find the probability that the first heads appears on the \(k\) th trial for \(k=1,2\), and 3 .

Suppose the lifetime of a printer is exponentially distributed with parameter \(\lambda=0.2 /\) year. (a) What is the expected lifetime? (b) The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5\). Find \(x_{m}\).

A random experiment consists of flipping a biased coin with probability \(0.3\) of heads until the first time heads appears. Find the probability that heads appears for the first time on the fifth trial.

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) If the life span of an organism is exponentially distributed, and if \(x_{m}=4\) years, what is the hazard-rate function?

A random experiment consists of rolling a fair die until the first time an even number appears. Find the probability that the first even number appears on the third trial.

A random experiment consists of rolling a fair die until the first time a five or a six appears. Find the probability that the first five or six appears on the \(k\) th trial for \(k=1,2, \ldots, 5\).

The hazard-rate function of an organism is given by $$ \lambda(x)=0.1+0.5 e^{0.02 x}, \quad x \geq 0 $$ where \(x\) is measured in days. (a) What is the probability that the organism will live less than 10 days? (b) What is the probability that the organism will live for another five days, given that it survived the first five days?

A random experiment consists of flipping a fair coin until the first time heads appears. Find the probability that the first heads appears after the third trial.

A random experiment consists of rolling a fair die until the first six appears. Find the probability that the first six appears after the seventh trial.

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) Use a graphing calculator to numerically approximate the median lifetime if the hazard-rate function is $$ \lambda(x)=0.5+0.1 e^{0.2 x}, \quad x \geq 0 $$

A random experiment consists of flipping a fair coin until the first time heads appears. Find the probability that the first heads appears within the first four trials.

A random experiment consists of rolling a fair die until the first time a 1 or a 2 appears. Find the probability that the first. 1 or 2 appears within the first five trials.

An urn contains 1 black and 14 white balls. Balls are drawn at random, one at a time, until the black ball is selected. Each ball is replaced before the next ball is drawn. Find the probability that at least 20 draws are needed.

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) Find the median lifetime if the hazard-rate function is $$ \lambda(x)=\left(4 \times 10^{-5}\right) x^{2.2}, \quad x \geq 0 $$

An urn contains 1 black and \(n-1\) white balls. Balls are drawn at random, one at a time, until the black ball is selected. Each ball is replaced before the next ball is drawn. Find the probability that at least \(n\) draws are needed. What happens as \(n \rightarrow \infty ?\)

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having died by age \(x_{m}\) is \(0.5 .\) Find the median lifetime if the hazard-rate function is $$ \lambda(x)=\left(3.7 \times 10^{-6}\right) x^{2.7}, \quad x \geq 0 $$