"Maura guesses at all 10 questions on a..." was answered by students like you

kaibaloveyou3cj

Answered question

2022-02-12

Maura guesses at all 10 questions on a true/false test.
a) How do you find the probability that she got seven correct?
b) How do you find the probability that she got at least half correct?

Answer & Explanation

Iacopelli5co

Beginner2022-02-13Added 15 answers

a) Suppose that p and q are probabilities. Maura makes an educated guess. Given that there is only a true/false option,
$p = \frac{1}{2}, q = \frac{1}{2}$
We use binomial theorem to solve where $X \geq 5$ .
$P (X \geq 5) = P (X = 5) + P (X = 6) + P (X = 7) + P (X = 8) + P (X = 9) + P (X = 10)$
$P (X \geq 5) = 10 C_{5} {(\frac{1}{2})}^{5} {(\frac{1}{5})}^{5} + 10 C_{6} {(\frac{1}{2})}^{6} {(\frac{1}{2})}^{4} + 10 C_{7} {(\frac{1}{2})}^{7} {(\frac{1}{2})}^{3}$
$+ 10 C_{8} {(\frac{1}{2})}^{8} {(\frac{1}{2})}^{2} + 10 C_{9} {(\frac{1}{2})}^{9} {(\frac{1}{2})}^{1} + 10 C_{10} {(\frac{1}{2})}^{10} {(\frac{1}{2})}^{0}$
$P (X \geq 5) = 252 \cdot {(\frac{1}{2})}^{10} + 210 \cdot {(\frac{1}{2})}^{10} + 120 \cdot {(\frac{1}{2})}^{10} + 45 \cdot {(\frac{1}{2})}^{10} + 10 \cdot {(\frac{1}{2})}^{10} + 1 \cdot {(\frac{1}{2})}^{10}$
$P (X \geq 5) = {(\frac{1}{2})}^{10} \cdot (252 + 210 + 120 + 45 + 10 + 1)$
$P (X \geq 5) = {(\frac{1}{2})}^{10} \cdot (638)$
$P (X \geq 5) = 0.6230$

traciaul6y

Beginner2022-02-14Added 14 answers

b) $P (X = 7), n = 10, p = 0.5$
$X \sim B i n o m i a l (n = 10, p = 0.5)$
$P (X = 7) = (\begin{matrix} n \\ x \end{matrix}) p^{x} (1 - p)^{n - x} = (\begin{matrix} 10 \\ 7 \end{matrix}) (0.5)^{7} (1 - 0.5)^{10 - 7} = 120 \cdot (0.5)^{7} (0.5)^{3} = 0.117188 \approx 0.117188 \approx 0.117$
$P (X = 7) = 0.117$

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