4.E: Discrete Random Variables (Exercises)

4.2: Probability Distributioins for Discrete Random Variables

Basic

Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully.
1. \[\beginx &-2 &0 &2 &4 \\ \hline P(x) &0.3 &0.5 &0.2 &0.1\\ \end\]
2. \[\beginx &0.5 &0.25 &0.25\\ \hline P(x) &-0.4 &0.6 &0.8\\ \end\]
3. \[\beginx &1.1 &2.5 &4.1 &4.6 &5.3\\ \hline P(x) &0.16 &0.14 &0.11 &0.27 &0.22\\ \end\]
1. \[\beginx &0 &1 &2 &3 &4\\ \hline P(x) &-0.25 &0.50 &0.35 &0.10 &0.30\\ \end\]
2. \[\beginx &1 &2 &3 \\ \hline P(x) &0.325 &0.406 &0.164 \\ \end\]
3. \[\beginx &25 &26 &27 &28 &29 \\ \hline P(x) &0.13 &0.27 &0.28 &0.18 &0.14 \\ \end\]
1. $P(80)$.
2. $P(X>80)$.
3. $P(X\leq 80)$.
4. The mean $\mu$ of $X$.
5. The variance $\sigma ^2$ of $X$.
6. The standard deviation $\sigma $ of $X$.
1. $P(18)$.
2. $P(X>18)$.
3. $P(X\leq 18)$.
4. The mean $\mu$ of $X$.
5. The variance $\sigma ^2$ of $X$.
6. The standard deviation $\sigma $ of $X$.
1. $P(5\leq X\leq 9)$.
2. $P(X\geq 7)$.
3. The mean $\mu$ of $X$. (For fair dice this number is $7$).
4. The standard deviation $\sigma $ of $X$. (For fair dice this number is about $2.415$).
Applications
1. Borachio works in an automotive tire factory. The number $X$ of sound but blemished tires that he produces on a random day has the probability distribution \[\begin
  x &2 &3 &4 &5 \\ \hline P(x) &0.48 &0.36 &0.12 &0.04\\ \end\]
  1. Find the probability that Borachio will produce more than three blemished tires tomorrow.
  2. Find the probability that Borachio will produce at most two blemished tires tomorrow.
  3. Compute the mean and standard deviation of $X$. Interpret the mean in the context of the problem.
  1. Find the probability that the next litter will produce five to seven live pups.
  2. Find the probability that the next litter will produce at least six live pups.
  3. Compute the mean and standard deviation of $X$. Interpret the mean in the context of the problem.
  1. Find the probability that no more than ten days will be lost next summer.
  2. Find the probability that from $8$ to $12$ days will be lost next summer.
  3. Find the probability that no days at all will be lost next summer.
  4. Compute the mean and standard deviation of $X$. Interpret the mean in the context of the problem.
  1. Construct the probability distribution of $X$.
  2. Compute the expected value $E(X)$ of $X$. Interpret its meaning.
  3. Compute the standard deviation $\sigma $ of $X$.
  1. Construct the probability distribution of $X$.
  2. Compute the expected value $E(X)$ of $X$. Interpret its meaning.
  3. Compute the standard deviation $\sigma $ of $X$.
  1. Construct the probability distribution of $X$. (Two entries in the table will contain $C$).
  2. Compute the expected value $E(X)$ of $X$.
  3. Determine the value $C$ must have in order for the company to break even on all such policies (that is, to average a net gain of zero per policy on such policies).
  4. Determine the value $C$ must have in order for the company to average a net gain of $\$250$ per policy on all such policies.
  1. Construct the probability distribution of $X$. (Two entries in the table will contain $C$).
  2. Compute the expected value $E(X)$ of $X$.
  3. Determine the value $C$ must have in order for the company to break even on all such policies (that is, to average a net gain of zero per policy on such policies).
  4. Determine the value $C$ must have in order for the company to average a net gain of $\$150$ per policy on all such policies.
  1. Construct the probability distribution of $X$.
  2. Compute the expected value $E(X)$ of $X$, and interpret its meaning in the context of the problem.
  3. Compute the standard deviation of $X$.
  1. Construct the probability distribution of $X$.
  2. Compute the expected value $E(X)$ of $X$, and explain why this game is not offered in a casino (where 0 is not considered even).
  3. Compute the standard deviation of $X$.
  1. Find the average time the bus takes to drive the length of its route.
  2. Find the standard deviation of the length of time the bus takes to drive the length of its route.
  Additional Exercises
  1. The number $X$ of nails in a randomly selected $1$-pound box has the probability distribution shown. Find the average number of nails per pound. \[\beginx &100 &101 &102 \\ \hline P(x) &0.01 &0.96 &0.03 \\ \end\]
  2. Three fair dice are rolled at once. Let $X$ denote the number of dice that land with the same number of dots on top as at least one other die. The probability distribution for $X$ is \[\beginx &0 &u &3 \\ \hline P(x) &p &\frac&\frac\\ \end\]
    1. Find the missing value $u$ of $X$.
    2. Find the missing probability $p$.
    3. Compute the mean of $X$.
    4. Compute the standard deviation of $X$.
    1. Construct the probability distribution for $X$.
    2. Compute the mean $\mu$ of $X$.
    3. Compute the standard deviation $\sigma $ of $X$.
    1. Construct the probability distribution for $X$.
    2. Compute the mean $\mu$ of $X$.
    3. Compute the standard deviation $\sigma $ of $X$.
    1. Construct the probability distribution for the number $X$ of defective units in such a sample. (A tree diagram is helpful).
    2. Find the probability that such a shipment will be accepted.
    1. What number of customers does Shylock most often see in the bank the moment he enters?
    2. What number of customers waiting in line does Shylock most often see the moment he enters?
    3. What is the average number of customers who are waiting in line the moment Shylock enters?
    1. Compute the mean revenue per night if the cover is not installed.
    2. Use the answer to (a) to compute the projected total revenue per $90$-night season if the cover is not installed.
    3. Compute the projected total revenue per season when the cover is in place. To do so assume that if the cover were in place the revenue each night of the season would be the same as the revenue on a clear night.
    4. Using the answers to (b) and (c), decide whether or not the additional cost of the installation of the cover will be recovered from the increased revenue over the first ten years. Will the owner have the cover installed?
    Answers
    1. no: the sum of the probabilities exceeds $1$
      
      no: a negative probability
      
      no: the sum of the probabilities is less than $1$
      
      $0.4$
      
      $0.1$
      
      $0.9$
      
      $79.15$
      
      $\sigma ^2=1.5275$
      
      $\sigma =1.2359$
      
      $0.6528$
      
      $0.7153$
      
      $\mu =7.8333$
      
      $\sigma ^2=5.4866$
      
      $\sigma =2.3424$
      
      $0.79$
      
      $0.60$
      
      $\mu =5.8$, $\sigma =1.2570$
      
      \[\beginx &-1 &999 &499 &99 \\ \hline P(x) &\frac&\frac&\frac&\frac\\ \end\]
      
      $-0.4$
      
      $17.8785$
      
      \[\beginx &C &C &-150,000 \\ \hline P(x) &0.9825 & &0.0175 \\ \end\]
      
      $C-2625$
      
      $C \geq 2625$
      
      $C \geq 2875$
      
      \[\beginx &-1 &1 \\ \hline P(x) &\frac&\frac\\ \end\]
      
      $E(X)=-0.0526$. In many bets the bettor sustains an average loss of about $5.25$ cents per bet.
      
      $0.9986$
      
      $43.54$
      
      $1.2046$
      
      \[\beginx &0 &1 &2 &3 &4 &5 \\ \hline P(x) &\frac&\frac&\frac&\frac&\frac&\frac\\ \end\]
      
      $1.9444$
      
      $1.4326$
      
      \[\beginx &0 &1 &2 \\ \hline P(x) &0.902 &0.096 &0.002 \\ \end\]
      
      $0.902$
      
      $2523.25$
      
      $227,092.5$
      
      $270,000$
      
      The owner will install the cover.
      
      4.3: The Binomial Distribution
      
      Basic
      
      Determine whether or not the random variable $X$ is a binomial random variable. If so, give the values of $n$ and$p$. If not, explain why not.
      
      $X$ is the number of dots on the top face of fair die that is rolled.
      
      $X$ is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck.
      
      $X$ is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which $0.02\%$ of all parts are defective.
      
      $X$ is the number of times the number of dots on the top face of a fair die is even in six rolls of the die.
      
      $X$ is the number of dice that show an even number of dots on the top face when six dice are rolled at once.
      
      $X$ is the number of black marbles in a sample of $5$ marbles drawn randomly and without replacement from a box that contains $25$ white marbles and $15$ black marbles.
      
      $X$ is the number of black marbles in a sample of $5$ marbles drawn randomly and with replacement from a box that contains $25$ white marbles and $15$ black marbles.
      
      $X$ is the number of voters in favor of proposed law in a sample $1,200$ randomly selected voters drawn from the entire electorate of a country in which $35\%$ of the voters favor the law.
      
      $X$ is the number of fish of a particular species, among the next ten landed by a commercial fishing boat, that are more than $13$ inches in length, when $17\%$ of all such fish exceed $13$ inches in length.
      
      $X$ is the number of coins that match at least one other coin when four coins are tossed at once.
      
      $P(11)$
      
      $P(9)$
      
      $P(0)$
      
      $P(13)$
      
      $P(14)$
      
      $P(4)$
      
      $P(0)$
      
      $P(20)$
      
      $P(X \leq 3)$
      
      $P(X \geq 3)$
      
      $P(3)$
      
      $P(0)$
      
      $P(5)$
      
      $P(X \leq 2)$
      
      $P(X \geq 2)$
      
      $P(2)$
      
      $P(0)$
      
      $P(5)$
      
      $n = 10, p = 0.25, P(X \leq 6)$
      
      $n = 10, p = 0.75, P(X \leq 6)$
      
      $n = 15, p = 0.75, P(X \leq 6)$
      
      $n = 15, p = 0.75, P(12)$
      
      $n = 15, p=0.\bar, P(10\leq X\leq 12)$
      
      $n = 5, p = 0.05, P(X \leq 1)$
      
      $n = 5, p = 0.5, P(X \leq 1)$
      
      $n = 10, p = 0.75, P(X \leq 5)$
      
      $n = 10, p = 0.75, P(12)$
      
      $n = 10, p=0.\bar, P(5\leq X\leq 8)$
      
      $n = 8, p = 0.43$
      
      $n = 47, p = 0.82$
      
      $n = 1200, p = 0.44$
      
      $n = 2100, p = 0.62$
      
      $n = 14, p = 0.55$
      
      $n = 83, p = 0.05$
      
      $n = 957, p = 0.35$
      
      $n = 1750, p = 0.79$
      
      $n = 5, p=0.\bar$
      
      $n = 10, p = 0.75$
      
      $n = 10, p = 0.25$
      
      $n = 15, p = 0.1$
      
      Find the probability that it lands heads up at most five times.
      
      Find the probability that it lands heads up more times than it lands tails up.
      
      Applications
      
      An English-speaking tourist visits a country in which $30\%$ of the population speaks English. He needs to ask someone directions.
      
      Find the probability that the first person he encounters will be able to speak English.
      
      The tourist sees four local people standing at a bus stop. Find the probability that at least one of them will be able to speak English.
      
      Find the probability that a carton of one dozen eggs contains no eggs that are either cracked or broken.
      
      Find the probability that a carton of one dozen eggs has (i) at least one that is either cracked or broken; (ii) at least two that are cracked or broken.
      
      Find the average number of cracked or broken eggs in one dozen cartons.
      
      Verify that $X$ satisfies the conditions for a binomial random variable, and find $n$ and $p$.
      
      Find the probability that $X$ is zero.
      
      Find the probability that $X$ is two, three, or four.
      
      Find the probability that $X$ is at least five.
      
      Find the average number of inferior quality grapefruit per box of a dozen.
      
      A box that contains two or more grapefruit of inferior quality will cause a strong adverse customer reaction. Find the probability that a box of one dozen grapefruit will contain two or more grapefruit of inferior quality.
      
      Find the probability that (i) none of the ten skeins will contain a knot; (ii) at most one will.
      
      Find the expected number of skeins that contain knots.
      
      Find the most likely number of skeins that contain knots.
      
      Verify that $X$ satisfies the conditions for a binomial random variable, and find $n$ and $p$.
      
      Find the probability that on any given day between five and nine patients will require a sedative (include five and nine).
      
      Find the average number of patients each day who require a sedative.
      
      Using the cumulative probability distribution for $X$ in 7.1: Large Sample Estimation of a Population Mean find the minimum number $x_$ of doses of the sedative that should be on hand at the start of the day so that there is a $99\%$ chance that the laboratory will not run out.
      
      Construct the probability distribution of $X$, the number of sales made each day.
      
      Find the probability that, on a randomly selected day, the salesman will make a sale.
      
      Assuming that the salesman makes $20$ sales calls per week, find the mean and standard deviation of the number of sales made per week.
      
      Additional Exercises
      
      When dropped on a hard surface a thumbtack lands with its sharp point touching the surface with probability $2/3$; it lands with its sharp point directed up into the air with probability $1/3$. The tack is dropped and its landing position observed $15$ times.
      
      Find the probability that it lands with its point in the air at least $7$ times.
      
      If the experiment of dropping the tack $15$ times is done repeatedly, what is the average number of times it lands with its point in the air?
      
      Find the probability that the proofreader will miss at least one of them.
      
      Show that two such proofreaders working independently have a $99.96\%$ chance of detecting an error in a piece of written work.
      
      Find the probability that two such proofreaders working independently will miss at least one error in a work that contains four errors.
      
      A student guesses the answer to every question. Find the chance that he guesses correctly between four and seven times.
      
      Find the minimum score the instructor can set so that the probability that a student will pass just by guessing is $20\%$ or less.
      
      If a carrier (not known to be such, of course) is boarded with three other dogs, what is the probability that at least one of the three healthy dogs will develop kennel cough?
      
      If a carrier is boarded with four other dogs, what is the probability that at least one of the four healthy dogs will develop kennel cough?
      
      The pattern evident from parts (a) and (b) is that if $K+1$ dogs are boarded together, one a carrier and $K$ healthy dogs, then the probability that at least one of the healthy dogs will develop kennel cough is $P(X\geq 1)=1-(0.992)^K$, where $X$ is the binomial random variable that counts the number of healthy dogs that develop the condition. Experiment with different values of $K$ in this formula to find the maximum number $K+1$ of dogs that a kennel owner can board together so that if one of the dogs has the condition, the chance that another dog will be infected is less than $0.05$.
      
      Show that the expected number of diseased individuals in the group of $600$ is $12$ individuals.
      
      Instead of testing all $600$ blood samples to find the expected $12$ diseased individuals, investigators group the samples into $60$ groups of $10$ each, mix a little of the blood from each of the $10$ samples in each group, and test each of the $60$ mixtures. Show that the probability that any such mixture will contain the blood of at least one diseased person, hence test positive, is about $0.18$.
      
      Based on the result in (b), show that the expected number of mixtures that test positive is about $11$. (Supposing that indeed $11$ of the $60$ mixtures test positive, then we know that none of the $490$ persons whose blood was in the remaining $49$ samples that tested negative has the disease. We have eliminated $490$ persons from our search while performing only $60$ tests.)
      
      Answers
      
      not binomial; not success/failure.
      
      not binomial; trials are not independent.
      
      binomial; $n = 10, p = 0.0002$
      
      binomial; $n = 6, p = 0.5$
      
      binomial; $n = 6, p = 0.5$
      
      $0.2434$
      
      $0.2151$
      
      $0.18^\approx 0$
      
      $0$
      
      $0.8125$
      
      $0.5000$
      
      $0.3125$
      
      $0.0313$
      
      $0.0312$
      
      $0.9965$
      
      $0.2241$
      
      $0.0042$
      
      $0.2252$
      
      $0.5390$
      
      $\mu = 3.44, \sigma = 1.4003$
      
      $\mu = 38.54, \sigma = 2.6339$
      
      $\mu = 528, \sigma = 17.1953$
      
      $\mu = 1302, \sigma = 22.2432$
      
      $\mu = 1.6667, \sigma = 1.0541$
      
      $\mu = 7.5, \sigma = 1.3693$
      
      $0.3$
      
      $0.7599$
      
      $n = 20, p = 0.1$
      
      $0.1216$
      
      $0.5651$
      
      $0.0432$
      
      $0.0563$ and $0.2440$
      
      $2.5$
      
      $2$
      
      $0.0776$
      
      $0.9996$
      
      $0.0016$
      
      $0.0238$
      
      $0.0316$
      
      $6$
      
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      4.3: The Binomial Distribution
      
      5: Continuous Random Variables
      
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4.E: Discrete Random Variables (Exercises)

4.2: Probability Distributioins for Discrete Random Variables

Basic

Applications

Additional Exercises

Answers

4.3: The Binomial Distribution

Basic

Applications

Additional Exercises

Answers

Contributor

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