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Hamid is playing a trivia game with multiple choice questions. Each question has 2 correct answers among 5 answer choices.
Hamid has no idea what the answers to a certain question are, so he needs to choose two different answers at random.
What is the probability that Hamid guesses both answers correctly?
Round your answer to two decimal places.

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Hamid is playing a trivia game with multiple choice questions. Each question has $2$ correct answers among $5$ answer choices. $\newline$ Hamid has no idea what the answers to a certain question are, so he needs to choose two different answers at random. $\newline$ What is the probability that Hamid guesses both answers correctly? $\newline$ Round your answer to two decimal places. $\newline$ $\square$

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Find probabilities using combinations and permutations

Full solution

Q. Hamid is playing a trivia game with multiple choice questions. Each question has

2

correct answers among

5

answer choices.

\newline

Hamid has no idea what the answers to a certain question are, so he needs to choose two different answers at random.

\newline

What is the probability that Hamid guesses both answers correctly?

\newline

Round your answer to two decimal places.

\newline

\square

Calculate Total Ways: Calculate the total number of ways to choose $2$ answers from $5$ choices. $\newline$ Use the combination formula: ${}_5C_2 = \frac{5!}{2!(5-2)!}$
Perform Calculation: Perform the calculation for $_5C_2$ . $_5C_2 = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10$
Choose Correct Answers: Calculate the total number of ways to choose the $2$ correct answers from the $2$ available. $\newline$ Since there are $2$ correct answers, the number of ways to choose them is simply $1$ (since you're choosing both).
Find Probability: Find the probability that Hamid guesses both answers correctly. $\newline$ Probability = $\frac{\text{Number of ways to choose the correct answers}}{\text{Total number of ways to choose any } 2 \text{ answers}}$
Perform Probability Calculation: Perform the calculation for the probability. $\newline$ Probability = $\frac{1}{10}$
Round Probability: Round the probability to two decimal places. $\newline$ Probability $\approx 0.10$

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