Chapter 12

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

In how many ways can four red and five black cards be selected from a standard deck of cards if cards are drawn without replacement?

A true-false exam has 20 questions. Find the expected number of correct answers if a student guesses the answers at random.

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. 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 repeat these steps 10 times. 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?

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?

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 cards) 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, \(\mathrm{Q} \mathrm{Q} \mathrm{Q} 33) ?\) (This is called a full house in poker.)

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 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?

Counterpoint 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?

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 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 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 six green, eight blue, and 10 red balls. You take one ball out of the urn, note its color, and replace it. You repeat these steps six times. 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 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.?

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?

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?

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

A number of human traits are caused by a single pair of recessive genes and thus manifest themselves only in individuals who are homozygous for 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. 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 human traits are caused by a single pair of recessive genes and thus manifest themselves only in individuals who are homozygous for 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 technical device is exponentially distributed with mean five years. (a) Find the probability that the device will have failed after three years. (b) Given that the device 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 technical device 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 failed 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 failed 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.

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having failed by age \(x_{m}\) is \(0.5\). Use a graphing calculator to numerically approximate the median lifetime if the hazard-rate function is $$ \lambda(x)=1.2+0.3 e^{0.5 x}, \quad x \geq 0 $$

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 failed 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.

The hazard-rate function of an organism is given by $$ \lambda(x)=0.04 x^{3.1}, \quad x \geq 0 $$ where \(x\) is measured in years. (a) What is the probability that the organism will live for more than three years? (b) What is the probability that the organism will live for another three years, given that it survived the first three years?

An urn contains one 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 failed 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 one 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 failed by age \(x_{m}\) is \(0.5 .\) Find the median lifetime if the hazard-rate function is $$ \lambda(x)=\left(3.7 \times 10^{-6}\right) x^{2.7}, \quad x \geq 0 $$

An urn contains five green and 25 blue balls. Balls are drawn at random, one at a time, until a green ball is selected. Each ball is replaced before the next ball is drawn. Let \(T\) denote the first time until a green ball is drawn. Find \(E(T)\) and \(\operatorname{var}(T)\).