Binomial Theorem - Algebra 2

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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?

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

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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 term is

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

To find the the coefficient of , 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

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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

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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?

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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:

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Question

Which is equivalent to ?

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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?

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

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Question

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

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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.

Tap to reveal answer

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.

Recall that:

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

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?

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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?

Tap to reveal answer

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

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?

Tap to reveal answer

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

Tap to reveal answer

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 term is

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

To find the the coefficient of , 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

Tap to reveal answer

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

← Didn't Know|Knew It →

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?

Tap to reveal answer

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:

← Didn't Know|Knew It →

Question

Which is equivalent to ?

Tap to reveal answer

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:

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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

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Question

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

Tap to reveal answer

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.

Tap to reveal answer

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.

Recall that:

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

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?

Tap to reveal answer

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?

Tap to reveal answer

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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