Math / Data & Statistics

QuestionTime left 0.0627
A multiple-choice test contains 24 questions, each with five answers. Assume a student just guesses on each question. what is the probability the student answers less than Four questions correctly? (Answer to the nearest three decimals 0.000).
Answer: $\square$ Next page

Studdy Solution

STEP 1

What is this asking? If a student randomly guesses on a 24-question multiple-choice test with five options for each question, what's the chance they get fewer than four questions right? Watch out! "Less than four" means three or fewer, not including four!
Also, remember each guess is independent.

STEP 2

1. Define the Binomial Probability
2. Calculate P(X=0)
3. Calculate P(X=1)
4. Calculate P(X=2)
5. Calculate P(X=3)
6. Sum the Probabilities

STEP 3

We're dealing with a binomial distribution here!
This is because each question is a separate trial with only two outcomes: right or wrong.
The probability of guessing correctly on any single question is $p = \frac{1}{5} = 0.2$ , since there are five choices.
The probability of guessing incorrectly is $1 - p = 1 - 0.2 = 0.8$ .
We want to find the probability of getting less than four questions correct, which means 0, 1, 2, or 3 correct.

STEP 4

The binomial probability formula is given by: $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ Where $n$ is the number of trials (24 questions), $k$ is the number of successes (correct answers), and $p$ is the probability of success on a single trial (0.2). $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$ .
It represents the number of ways to choose $k$ successes from $n$ trials.

STEP 5

Let's start with zero correct answers ( $k=0$ ): $P(X=0) = \binom{24}{0} \cdot (0.2)^0 \cdot (0.8)^{24-0} = 1 \cdot 1 \cdot (0.8)^{24} \approx \mathbf{0.0047}$

STEP 6

Now, let's find the probability of getting exactly one question right ( $k=1$ ): $P(X=1) = \binom{24}{1} \cdot (0.2)^1 \cdot (0.8)^{24-1} = 24 \cdot 0.2 \cdot (0.8)^{23} \approx \mathbf{0.0274}$

STEP 7

Next, the probability of getting exactly two questions right ( $k=2$ ): $P(X=2) = \binom{24}{2} \cdot (0.2)^2 \cdot (0.8)^{24-2} = 276 \cdot 0.04 \cdot (0.8)^{22} \approx \mathbf{0.0755}$

STEP 8

Finally, the probability of getting exactly three questions right ( $k=3$ ): $P(X=3) = \binom{24}{3} \cdot (0.2)^3 \cdot (0.8)^{24-3} = 2024 \cdot 0.008 \cdot (0.8)^{21} \approx \mathbf{0.1367}$

STEP 9

To find the probability of getting less than four correct, we add the probabilities of getting 0, 1, 2, or 3 correct: $P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)$ $P(X<4) \approx 0.0047 + 0.0274 + 0.0755 + 0.1367 \approx \mathbf{0.2443}$

STEP 10

The probability of answering fewer than four questions correctly is approximately $\mathbf{0.244}$ .

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QuestionTime left 0.0627
A multiple-choice test contains 24 questions, each with five answers. Assume a student just guesses on each question. what is the probability the student answers less than Four questions correctly? (Answer to the nearest three decimals 0.000).
Answer: $\square$ Next page

Studdy Solution

STEP 1

What is this asking? If a student randomly guesses on a 24-question multiple-choice test with five options for each question, what's the chance they get fewer than four questions right? Watch out! "Less than four" means three or fewer, not including four!
Also, remember each guess is independent.

STEP 2

1. Define the Binomial Probability
2. Calculate P(X=0)
3. Calculate P(X=1)
4. Calculate P(X=2)
5. Calculate P(X=3)
6. Sum the Probabilities

STEP 3

STEP 4

STEP 5

Let's start with zero correct answers ( $k=0$ ): $P(X=0) = \binom{24}{0} \cdot (0.2)^0 \cdot (0.8)^{24-0} = 1 \cdot 1 \cdot (0.8)^{24} \approx \mathbf{0.0047}$

STEP 6

Now, let's find the probability of getting exactly one question right ( $k=1$ ): $P(X=1) = \binom{24}{1} \cdot (0.2)^1 \cdot (0.8)^{24-1} = 24 \cdot 0.2 \cdot (0.8)^{23} \approx \mathbf{0.0274}$

STEP 7

Next, the probability of getting exactly two questions right ( $k=2$ ): $P(X=2) = \binom{24}{2} \cdot (0.2)^2 \cdot (0.8)^{24-2} = 276 \cdot 0.04 \cdot (0.8)^{22} \approx \mathbf{0.0755}$

STEP 8

Finally, the probability of getting exactly three questions right ( $k=3$ ): $P(X=3) = \binom{24}{3} \cdot (0.2)^3 \cdot (0.8)^{24-3} = 2024 \cdot 0.008 \cdot (0.8)^{21} \approx \mathbf{0.1367}$

STEP 9

To find the probability of getting less than four correct, we add the probabilities of getting 0, 1, 2, or 3 correct: $P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)$ $P(X<4) \approx 0.0047 + 0.0274 + 0.0755 + 0.1367 \approx \mathbf{0.2443}$

STEP 10

The probability of answering fewer than four questions correctly is approximately $\mathbf{0.244}$ .

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QuestionTime left 0.0627A multiple-choice test contains 24 questions, each with five answers. Assume a student just guesses on each question. what is the probability the student answers less than Four questions correctly? (Answer to the nearest three decimals 0.000).Answer: □\square□ Next page

Studdy Solution

STEP 1

STEP 2

1. Define the Binomial Probability2. Calculate P(X=0)3. Calculate P(X=1)4. Calculate P(X=2)5. Calculate P(X=3)6. Sum the Probabilities

STEP 3

STEP 4

STEP 5

Let's start with zero correct answers (k=0k=0k=0): P(X=0)=(240)⋅(0.2)0⋅(0.8)24−0=1⋅1⋅(0.8)24≈0.0047P(X=0) = \binom{24}{0} \cdot (0.2)^0 \cdot (0.8)^{24-0} = 1 \cdot 1 \cdot (0.8)^{24} \approx \mathbf{0.0047}P(X=0)=(024​)⋅(0.2)0⋅(0.8)24−0=1⋅1⋅(0.8)24≈0.0047

STEP 6

STEP 7

STEP 8

STEP 9

STEP 10

The probability of answering fewer than four questions correctly is approximately 0.244\mathbf{0.244}0.244.

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QuestionTime left 0.0627
A multiple-choice test contains 24 questions, each with five answers. Assume a student just guesses on each question. what is the probability the student answers less than Four questions correctly? (Answer to the nearest three decimals 0.000).
Answer: $\square$ Next page

1. Define the Binomial Probability
2. Calculate P(X=0)
3. Calculate P(X=1)
4. Calculate P(X=2)
5. Calculate P(X=3)
6. Sum the Probabilities

Let's start with zero correct answers ( $k=0$ ): $P(X=0) = \binom{24}{0} \cdot (0.2)^0 \cdot (0.8)^{24-0} = 1 \cdot 1 \cdot (0.8)^{24} \approx \mathbf{0.0047}$

The probability of answering fewer than four questions correctly is approximately $\mathbf{0.244}$ .