Binomial Distribution

Posted November 12th, 2009 by babygurl321

Probability and Statistics questions

a Studnet takes a 20 question multiple-choice exam with five choices for each question and guesses on each question. Find the probability of guessing at least 15 out of 20 correctly. Would you consider this event likely or unlikey to occur? Explain

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On November 14th, 2009 Structure says:

$X_n=\left [\begin{array}{cccccccc} 0&1&2&.&k&.&n-1&n\\ \left (\begin{array}{c}n\\0 \end {array}\right ) \)q^n&\left (\begin{array}{c}n\\1 \end {array}\right )\)pq^{n-1}&\left (\begin{array}{c}n\\2 \end {array}\right ) \)p^2q^{n-2}&.&\left (\begin{array}{c}n\\k \end {array}\right ) \)p^kq^{n-k}&.&\left (\begin{array}{c}n\\n-1 \end {array}\right ) \)p^{n-1}q^&\left (\begin{array}{c}n\\n \end {array}\right ) \)p^n \end {array}\right ] \)$

$X_{20}=\left [\begin{array}{cccccccc} 0&1&2&.&k&.&n-1&n\\ \left (\begin{array}{c}20\\0 \end {array}\right ) \)q^{20}&\left (\begin{array}{c}20\\1 \end {array}\right )\)pq^{19}&\left (\begin{array}{c}20\\2 \end {array}\right ) \)p^2q^{18}&.&\left (\begin{array}{c}20\\k \end {array}\right ) \)p^kq^{20-k}&.&\left (\begin{array}{c}20\\19 \end {array}\right ) \)p^{19}q^&\left (\begin{array}{c}20\\20 \end {array}\right ) \)p^{20} \end {array}\right ] \)$
In your case error occurs with probability

$q=\frac{4}{5}$

You guess with probability

$p=\frac{1}{5}$

You want to guess at least 15, so you have to sum up probabilities from 15 to 20.

$prob=\sum_{k=15}^{k=20}\left (\begin{array}{c}20\\k \end {array}\right ) \)(\frac{1}{5})^k(\frac{4}{5})^{20-k}=\frac{1}{5^{20}}\sum_{k=15}^{k=20}\left (\begin{array}{c}20\\k \end {array}\right ) \)4^{20-k}=\frac{1}{5^{20}}\sum_{k=0}^{k=5}\left (\begin{array}{c}20\\k \end {array}\right ) \)4^{k}$