Probability - Algebra II

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Question

Assume that you guess on each question of a multiple choice test. There are 12 questions and each question has 4 possible answers. What is the probability of getting exactly 8 answers correct?

Answer

This problem can be solved using the binomial probability equation:

$\small P=\binom{n}{k}p^kq^{n-k}$

$\small n=total\ number\ of\ questions\ (trials)$

$\small k=number\ correct\ (successes)$

$\small n-k=number\ incorrect\ (failures)$

$\small p=probability\ of\ getting\ 1\ question\ correct\ (0.25)$

$\small q=1-p\ probability\ of\ getting\ 1\ question\ incorrect\ (0.75)$

$\small \small \binom{n}{k}=\frac{n!}{k!(n-k)!}$

$\small \small P=(\frac{n!}{k!(n-k)!})p^kq^{n-k}=\frac{12!}{8!(4)!}(0.25)^8(0.75)^4$

$\small P=\frac{12\times11\times10\times9}{4\times3\times2\times1}(0.25)^8(0.75)^4=0.002$

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Question

What is the coefficient of $x^{6}$ if the expression $(x + 4)^{8}$ is expanded?

Answer

By the Binomial Theorem, if the expression $(x+a) ^{n}$ is expanded, the result can be defined as

$(a+b) ^{n} = \sum_{i=0}^{n} C(n, i) a^{n-i} b^{i}$

If we set , then the above expression, with slight rearranging, becomes

$(x+4) ^{8} = \sum_{i=0}^{8} C(8, i) \cdot 4^{i} \cdot x^{8-i}$

The coefficient of the $x^{8-i}$ term is

$C(8, i) \cdot 4^{i}$

To find the the coefficient of $x^{6}$ , we set and evaluate:

$C(8, 2) \cdot 4^{2} = 28 \cdot 16 = 448$

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Question

What is the coefficient of in the polynomial

Answer

The binomial theorem says that we can represent the polynomial as a sum

$\sum_{k=0}^n\binom{n}{k} a^k x^{n-k}$

Thus, if we want to find the coefficient of ,

since ,

and .

Therefore the coefficient of is

$\binom{7}{4}3^3 = \frac{7!}{4!(7-4)!}3^3 = 945$

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Question

You're taking a multiple choice quiz that has questions. If each question has choices and you guess on all of them, what is the probability of getting exactly questions correct?

Answer

This question requires the application of the binomial theorem for probability. In order to determine the probability of getting exactly 6 questions right, we must remember the formula for this theorem:

$P=\begin{pmatrix} n\k \end{pmatrix}p^kq^{n-k}$

Where is the number of trials (total questions), is the number of successes (correct answers), is the probability of success in one trial (chance of answering a question correctly), is the probability of failure in one trial (chance of answering a question incorrectly), and is the probability of getting questions correct out of total questions. Because there are 5 choices for each question, the chances of answering a question correctly are 1/5, or 0.2, and therefore the chances of answering a question incorrectly are 4/5, or 0.8. Now we have all of our values and can plug them into the formula:

$P=\begin{pmatrix} 10\6 \end{pmatrix}(0.2)^6(0.8)^{10-6}$

The first part of the formula in which we have a 10 over a 6 in parentheses means we perform the following calculation:

$\begin{pmatrix} 10\6 \end{pmatrix}=\frac{10!}{(10-6)!6!}=\frac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{(4\cdot3\cdot2\cdot1)(6\cdot5\cdot4\cdot3\cdot2\cdot1)}$

$=\frac{10\cdot9\cdot8\cdot7}{4\cdot3\cdot2\cdot1}=\frac{5040}{24}=210$

So now we can put this value into our formula, which gives us:

$P=(210)(0.2)^6(0.8)^{10-6}=0.0055$

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Question

Which is equivalent to $(a+b)^{4}$ ?

Answer

To answer this question, you could either use the binomial theorem or multiply . Both are detailed below.

Note: If you need help understanding Sigma Notation, please visit: https://www.varsitytutors.com/hotmath/hotmath_help/topics/sigma-notation-of-a-series; additionally, the notation is often read out loud as “n choose k” and is another way to write For more information on combinations, visit https://www.varsitytutors.com/hotmath/hotmath_help/topics/combinations

The Binomial Theorem is:

In our case, n=4. Plugging this in, we get:

Alternatively, we can solve this problem by multiplying to get the following:

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Question

A fair coin is tossed 10 times. What is the probability that four heads will be observed?

Answer

This is a binomial distribution with number of trials(n) equal to 10. The probability of success(p) is 0.5 because it is a fair coin. The number of success(r) is 4 because we want the probability of 4 heads.

The formula for a binomial distribution is

$(_nC_r)p^r(1-p)^{n-r}$

$(_{10}C_{4})*0.5^4+(1-0.5)^6$

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Question

A fair coin is tossed 50 times. What is the expected number of heads?

Answer

The expected probably for the binomial distribution is n*p. N is the number of trials, in this case the 50 coin tosses. p is the probability of heads. Since the coin is a fair coin the probability is .

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Question

A student takes a 12 question multiple choice test. There are 5 answer choices, and the student guesses on all the questions. What is the probability the student will get exactly 7 questions correct in order to pass? Round to 5 decimal places.

Answer

In order to determine the probability, we will need to use the binomial theorem.

The equation can be written in two ways:

$\binom{n}{r}\cdot p^r\cdot (1-p)^{n-r}$

Or:

$_nC_r \cdot p^r \cdot q^{n-r}$

Identify the definition and values for .

: represents the total number of trials
: represents the number of events
: represents the probability of occurrence per trial
: is the probability that the occurrence will not happen per trial

, , $p= \frac{1}{5}$ , $q = \frac{4}{5}$

Substitute the values into the formula.

$_{12}C_{7} \cdot (\frac{1}{5})^7 \cdot (\frac{4}{5})^{12-7}$

Recall that:

$C(n,r) =\frac{n!}{r! (n - r)!}$

$_{12}C_{7} = \frac{12!}{7!(12-7)!} = 792$

The terms become:

$792 \cdot (\frac{1}{5})^7 \cdot (\frac{4}{5})^{12-7} = \frac{811008}{244140625}$

Simplify the terms with calculator.

The answer is:

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Question

Suppose Billy takes a 5 question multiple choice test that has 5 answer choices per question. What is the probability that Billy will get exactly four correct to pass the test?

Answer

Write the binomial formula.

$P = _nC_r\cdot p^r\cdot q^{n-r} = \frac{n!}{r!(n-r)!}\cdot p^r\cdot q^{n-r}$

$n=\textup{number of trials }= 5$

$r= \textup{number of success} =4$

$p=\textup{probability of success per trial} = \frac{1}{5}$

$q=\textup{probability of failure per trail} = \frac{4}{5}$

Evaluate the probability.

$P=\frac{5!}{4!(5-4)!}\cdot (\frac{1}{5})^4\cdot (\frac{4}{5})^{5-4}$

$P = 5\cdot(\frac{1}{5})\cdot(\frac{1}{5})\cdot(\frac{1}{5})\cdot(\frac{1}{5})\cdot(\frac{4}{5})$

The answer is: $\frac{4}{625}$

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Question

Suppose a competitor has to answer four out of six multiple choice questions correctly to win a prize. There are four answer choices per question. What is the probability that this competitor will succeed answering exactly four correct questions if he or she guessed on all the questions?

Answer

This problem requires the binomial theorem. Write the formula.

$P = _nC_r\cdot p^r\cdot q^{n-r}$

This formula can also be rewritten as:

$P= \frac{n!}{r!(n-r)!}\cdot p^r\cdot q^{n-r}$

Identify all the terms.

$n=\textup{number of trials } =6$

$r= \textup{number of success} =4$

There are four answer choices per question, which means there is only one correct answer.

$p=\textup{probability of success per trial} = \frac{1}{4}$

The failure rate would be three out of the four questions.

$q=\textup{probability of failure per trail} = \frac{3}{4}$

Substitute the terms into the formula and simplify the terms.

$P= \frac{6!}{4!(6-4)!}\cdot (\frac{1}{4})^4\cdot (\frac{3}{4})^{6-4}$

$P= \frac{6\times 5}{2}\cdot (\frac{1}{4})(\frac{1}{4})(\frac{1}{4})(\frac{1}{4})\cdot (\frac{3}{4})(\frac{3}{4})$

$P= 15\cdot \frac{9}{4^6} = \frac{135}{4096}$

The answer is: $\frac{135}{4096}$

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Question

Krystal owns a boutique clothing store, and wants to learn more about her customers. She sends out a survey to the 35 customers who spent the most money in her store last month. Is this sample of customers likely to be biased?

Answer

Yes, this sample is likely to be biased. If Krystal only reaches out to the customers who spent the most money, she may fail to understand all types of consumer habits.

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Question

A marketer is piloting a set of new advertisement slogans--a potential replacement for her company’s old slogans--to determine the new slogans’ effectiveness at raising the number of clicks her ads receive. She partners with a social media company to test the slogans by targeting 1,000 random social media users each day with the new slogans and targeting 1,000 different random users each day with the old slogans.

Which of the following is the experimental group?

Answer

The experimental group in an experiment receives the experimental treatment. The dependent variable associated with the experimental group is affected by the independent variable in the experiment, while the dependent variable associated with the control group is not.

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Question

Miko is a psychologist conducting a study about biased grading. She wants to see whether English teachers will give a paper a higher grade if they have just graded a poorly written paper.

Miko recruits 30 teachers for the study and randomly sorts them into two groups. She has the 115 teachers in one group grade a poorly written paper, and the other 15 teachers grade an average paper. Afterward, all 30 teachers are given the same paper to grade. Miko records the grades for this paper.

Is this experiment statistically well designed?

Answer

Yes, this experiment is statistically well designed. In order to be statistically well designed, an experiment's subjects must be randomly sorted into two groups (experimental and control), everyone in the experimental group gets the experiment, and everyone in the control group does not get the experiment. Miko randomly sorted the teachers. Fifteen of them saw a poor paper (experiment) and fifteen saw an average paper (control). Afterwards, they were all given an identical paper to grade. While we don't know what the quality of that final identical paper was, the quality itself does not matter, but rather, how each of the 30 individuals grade it, and whether those in the experimental group do so differently than those in the control group IS what matters. Because the experiment has an experimental group and a control group, and both groups were randomly chosen, the experiment is statistically well designed.

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Question

Medical data shows it can take more than 4 months to recover from knee replacement surgery. A team of researchers is conducting an experiment to see if a new rehabilitation program can help reduce this time. The researchers ask 80 randomly chosen knee replacement patients to use the new program. They then note the recovery time for each patient.

Is this a statistically well designed experiment?

Answer

No, this experiment is not statistically well designed. In order to be statistically well designed, an experiment's subjects must be randomly sorted into two groups (experimental and control), everyone in the experimental group gets the experiment, and everyone in the control group does not get the experiment. All 80 participants were asked only to participate in the experimental trial; no one was placed into a control group. For this reason, this experiment is not statistically well designed.

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Question

A student wants to find out whether mothers in his city support a local initiative to redesign the city’s largest park. He decides to conduct a poll on this question by calling the home phone numbers of all 200 students in his private high school, asking to speak to their mothers, and recording their responses as YES, NO or NO PREFERENCE in a chart. He doesn’t have time to make 400 calls, so he enlists three friends, who are also students at his school, to help make calls and record answers.

100 calls are not answered, and during 50 calls no mothers are home. The remaining 250 calls successfully reach mothers: 200 of the 250 calls are recorded as YES, 25 are recorded as NO and 25 are recorded as NO PREFERENCE. The student concludes that most mothers in the city support the initiative.

Which of the following best describes whether this study is well-designed, and why?

Answer

Given the small population of students at the high school, it is highly unlikely that all mothers in the city are represented in the sample; and given that the high school is both small and private, it is unlikely that the sample is representative of all mothers in the city, creating sampling bias.

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Question

A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment.

Should the other group receive a placebo treatment?

Answer

It is well documented that when people believe they are receiving a treatment, they experience what they expect will be the effects of that treatment. Placebos help to verify whether an experimental treatment is effective by reducing the likelihood that study participants’ expectations or beliefs about treatments, or about whether they received a treatment, skew study results.

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Question

A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment, while the other receives a placebo treatment. No study participant is told whether he or she has received the experimental treatment or the placebo, but the researcher keeps careful track of which participants have received the experimental treatment and which have not.

What is the design of this study?

Answer

It is well documented that when people believe they are receiving a treatment, they experience what they expect will be the effects of that treatment. In a blind study, no participant being studied is aware which group he or she is in, or exactly what treatment he or she received. This can help prevent study participants’ expectations and beliefs about treatments, or about whether they received a treatment, from skewing study results.

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Question

A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment, while the other receives a placebo treatment. During the study, neither the researcher nor the study participants know who has received the experimental treatment, and who has received the placebo.

What is the design of this study?

Answer

When researchers are aware which study participants have received which treatments, they may consciously or subconsciously behave in ways that skew results of the study towards the researchers’ own perspectives or interests. In a double-blind study, during the study neither the researchers nor any participant is aware which group he or she is in, or exactly what treatment he or she received. This can prevent researcher behavior from skewing study results.

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Question

A marketer is piloting a set of new advertisement slogans--a potential replacement for her company’s old slogans--to determine the new slogans’ effectiveness at raising the number of clicks her ads receive. She partners with a social media company to test the slogans by targeting 1,000 random social media users each day with the new slogans and targeting 1,000 different random users each day with the old slogans.

Which of the following is the control group?

Answer

The control group in an experiment does not receive the experimental treatment, but resembles the group receiving the experimental treatment. Thus, the dependent variable associated with the control group is not affected by the independent variable in the experiment. This allows the researcher to compare the effect of the experimental treatment on the experimental group with the effect of no treatment, making the effect of the treatment easier to verify.

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Question

A marketer is piloting a set of new advertisement slogans--a potential replacement for her company’s old slogans--to determine the new slogans’ effectiveness at raising the number of clicks her ads receive. She partners with a social media company to test the slogans by targeting 1,000 random social media users each day with the new slogans and targeting 1,000 different random users each day with the old slogans. When the study ends, she compares the click activity on ads with each type of slogan and concludes that the new slogans are effective.

The results are represented in the table below:

Is her conclusion correct?

Answer

A statistically significant result of an experiment is one in which a change in the experimental group’s dependent variable is attributable to a change in the independent variable. In other words, a statistically significant result in a well designed experiment is one that is large enough in scale that it is unlikely to be a result of mere chance. In the table provided, the difference between the total ad clicks resulting from each of the slogan types is minuscule--a difference of only 3 clicks out of over 2,000 for each type. This suggests that the results may simply have been the result of chance, since the difference in total clicks is so small in scale that it is difficult to clearly attribute to the change in slogan types.

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