Chapter Questions
Find the probabilities that $\mathrm{X}$, the number of "successes" of a binomial experiment with 4 independent trials and $\pi=$ probability of "success" $=1 / 3$, equals $0,1,2,3$, or 4 . Sketch a histogram for the distribution.
Calculate the binomial probability distribution for $\mathrm{n}=10$ and $\mathrm{p}=1 / 8$. Compare this distribution with the distribution for $\mathrm{n}=10$ and $\mathrm{p}=1 / 2$. Do they differ in degree of dispersion? Position of peak? Maximum height of peak? Degree of symmetry?
John Reeves completed $60 \%$ of his passes one season. Assuming he is as good a quarterback the next fall, what is the probability that he will complete 80 of his first 100 passes?
Expand $(\mathrm{x}+2 \mathrm{y})^{5}$
Find the expansion of $(\mathrm{x}+\mathrm{y})^{6}$
Give the expansion of $\left[\mathrm{x}^{2}-(1 / \mathrm{s})\right]^{5}$.
Thirteen machines are in operation. The probability that at the end of one day a machine is still in operation is $0.6 .$ If the machines function independently, find the most probable number of machines in operation at the end of that day and the probability that these many machines are operating.
What is the probability of getting exactly 3 heads in 5 flips of a balanced coin?
On three successive flips of a fair coin, what is the probability of observing 3 heads? 3 tails ?
Find the probability that in three rolls of a pair of dice, exactly one total of 7 is rolled.
What is the probability of getting exactly 4 "sixes" when a die is rolled 7 times?
A deck of cards can be dichotomized into black cards and red cards. If $\mathrm{p}$ is the probability of a black card on a single draw and $\mathrm{q}$ the probability of a red card, $\mathrm{p}=1 / 2$ and $\mathrm{q}=1 / 2$. Six cards are sampled with replacement. What is the probability on six draws of getting 4 black and 2 red cards? Of getting all black cards?
In a family of 4 children, what is the probability that there will be exactly two boys?
Suppose that the probability of parents to have a child with blond hair is $1 / 4$. If there are four children in the family, what is the probability that exactly half of them have blond hair?
If a fair coin is tossed four times, what is the probability of at least two heads?
A baseball player has a $.250$ batting average (one base hit every four times, on the average). Assuming that the binomial distribution is applicable, if he is at bat four times on a particular day, what is (a) the probability that he will get exactly one hit? (b) the probability that he will get at least one hit?
If a deck of cards is dichotomized into hearts and all other cards, what is the probability $\mathrm{p}$ of getting a heart on a single draw? What is the probability q of getting a spade, club, or diamond? When 7 cards are sampled with replacement, what is the probability of getting no hearts at all? What is the probability of getting 4 hearts? What is the probability of getting 2 hearts out of the first 4 draws and then 2 hearts out of the next $3 ?$ Is this result more or less probable than "4 hearts out of $7^{\prime \prime} ?$ Why?
If the probability of your hitting a target on a single shot is .8, what is the probability that in four shots you will hit the target at least twice?
Records of an insurance company show that $3 / 1000$ of the accidents reported to the company involve a fatality. Determine: (a) the probability that no fatality is involved in thirty accidents reported (b) the probability that four fatal accidents are included in twenty accidents reported.
Given that $40 \%$ of entering college students do not complete their degree programs, what is the probability that out of 6 randomly selected students, more than half will get their degrees?
The probability that a basketball player makes at least one of six free throws is equal to $0.999936 .$ Find: (a) the probability function of $\mathrm{X}$, the number of times he scores; (b) the probability that he makes at least three baskets.
If 40 percent of a company's employees are in favor of a proposed new incentive-pay system, develop the probability distribution for the number of employees out of a sample of two employees who would be in favor of the incentive system by the use of a tree diagram. Use $F$ for a favorable reaction and $\mathrm{F}^{\prime}$ for an unfavorable reaction.
Over a long period of time a certain drug has been effective in 30 percent of the cases in which it has been prescribed. If a doctor is now administering this drug to four patients, what is the probability that it will be effective for at least three of the patients?
You are told that 9 out of 10 doctors recommend Potter's Pills. Assuming this is true, suppose you plan to choose 4 doctors at random. What is the probability that no more than two of these 4 doctors will recommend Potter's Pills?
The most common application of the binomial theorem in industrial work is in lot-by-lot acceptance $\quad$ inspection. If there are a certain number of defectives in the lot, the lot will be rejected as unsatisfactory. It is natural to wish to find the probability that the lot is acceptable even though a certain number of defectives are observed. Let $\mathrm{p}$ be the fraction of defectives in the lot. Assume that the size of the sample is small compared to the lot size. This will insure that the probability of selecting a defective item remains constant from trial to trial. Now choose a sample of size 18 from a lot where $10 \%$ of the items are defective. What is the probability of observing 0,1 or 2 defectives in the sample.
Over a period of some years, a car manufacturing firm finds that $18 \%$ of their cars develop body squeaks within the guarantee period. In a randomly selected shipment, 20 cars reach the end of the guarantee period and none develop squeaks. What is the probability of this?
A sample of 4 fuses is selected without replacement from a lot consisting of 5000 fuses. Assuming that $20 \%$ of the fuses in the lot are known to be defective, what is the probability that the sample would contain exactly 2 defective items?
An industrial process produces items of which $1 \%$ are defective. If a random sample of 100 of these are drawn from a large consignment, calculate the probability that the sample contains no defectives.
A proportion $\mathrm{p}$ of a large number of items in a batch is defective. A sample of n items is drawn and if it contains no defective items the batch is accepted while if it contains more than two defective items the batch is rejected. If, on the other hand, it contains one or two defectives, an independent sample of $\mathrm{m}$ is drawn, and if the combined number of defectives in the samples does not exceed two, the batch is accepted. Calculate the possibility of accepting this batch.
Let $\mathrm{X}$ be a binomially distributed random variable with parameters $\mathrm{n}$ and $\pi .$ where $\mathrm{n}=$ the number of independent trials and $\pi$ is the probability of success on a particular trial. Use the table above to find $\operatorname{Pr}(\mathrm{X} \geq 2)$ and $\operatorname{Pr}(\mathrm{X}=2)$if $\mathrm{n}=4$ and $\pi=.23$.
For $\mathrm{n}=4$ and $\pi=0.73$, find (a) $\mathrm{P}(\mathrm{X} \leq 2)$ and (b) $\mathrm{P}(\mathrm{X}=2)$.
Given the following cumulative binomial distribution, find (a) $\mathrm{P}(\mathrm{X}=1) ;$ (b) $\mathrm{P}(\mathrm{X}=4) ;$ (c) $\mathrm{P}(\mathrm{X}=5)$.$$(\mathrm{n}=5, \mathrm{p}=0.31)$$
Given the following cumulative binomial distribution, find (a) $\mathrm{P}(\mathrm{X}=4)$; (b) $\mathrm{P}(\mathrm{X}=1)$; (c) $\mathrm{P}(\mathrm{X}=0)$.$$(\mathrm{n}=5, \mathrm{p}=0.69)$$
The probability of hitting a target on a shot is $2 / 3$. If a person fires 8 shots at a target, Let $X$ denote the number of times he hits the target, and find:(a) $\mathrm{P}(\mathrm{X}=3)$ (b) $\mathrm{P}(1<\mathrm{X} \leq 6)$(c) $\mathrm{P}(\mathrm{X}>3)$.
If a bag contains three white two black, and four red balls and four balls are drawn at random with replacement, calculate the probabilities that(a) The sample contains just one white ball.(b) The sample contains just one white ball given that it contains just one red ball.
Three electric motors from a factory are tested. A motor is either discarded, returned to the factory or accepted. If the probability of acceptance is $.7$, the probability of return is $.2$ and the probability of discard is $.1$, what is the probability that of three randomly selected motors 1 will be returned 1 will be accepted and 1 will be discarded? What is the probability that 2 motors will be accepted, 1 returned and 0 discarded?
A survey was made of the number of people who read classified ads in a newspaper. Thirty people were asked to indicate which one of the following best applies to them:(1) read no ads (N); (2) read "articles for sale" ads (S);(3) read "help wanted" ads $(\mathrm{H}) ;$ (4) read all ads (A).(a) Use the multinomial theorem for the expansion of$$(\mathrm{N}+\mathrm{S}+\mathrm{H}+\mathrm{A})^{30}$$to find the coeffients of the terms involving(1)$$\mathrm{N}^{10} \mathrm{~A}^{10} \mathrm{H}^{10}$$(2) $\mathrm{N}^{5} \mathrm{~S}^{10} \mathrm{H}^{10} \mathrm{~A}^{5}$(b) Assuming the following probabilities, what is the probability that 10 read no ads, 10 read "Help Wanted" ads, and 10 read all ads?$$\begin{aligned}&\mathrm{P}\{\mathrm{N}\}=30 / 100 \\&\mathrm{P}\{\mathrm{S}\}=40 / 100 \\&\mathrm{P}\{\mathrm{H}\}=20 / 100 \\&\mathrm{P}\{\mathrm{A}\}=10 / 100\end{aligned}$$
Find the coefficient of $\mathrm{a}^{2} 1 \mathrm{a}_{2} \mathrm{a}_{3}$ in the expansion of$$\left(a_{1}+a_{2}+a_{3}\right)^{4}$$
A package in the mail can either be lost, delivered or damaged while being delivered. If the probability of loss is $.2$, the probability of damage is $.1$ and the probability of delivery is $.7$ and 10 packages are sent to Galveston, Texas, what is the probability that 6 arrive safely 2 are lost and 2 are damaged?
A die is tossed 12 times. Let $X$ : denote the number of tosses in which i dots come up for $\mathrm{i}=1,2,3,4,5$, and $6 .$ What is the probability that we obtain two of each value.