Question 1133310

CORRECTED answer below -- search for CORRECTED

The probability of getting exactly n correct is:
 {{{ P(n_corr) = (1/4)^(n)*(3/4)^(25-n) }}}<br>

So we can build this table:<br>

perl -e 'for($c=0; $c<26; $c++) { $p = (1/4)**($c)*(3/4)**(25-$c); printf "Pr(%d correct) = %e\n", $c, $p; }' 
Pr(0 correct) = 7.525435e-04
Pr(1 correct) = 2.508478e-04
Pr(2 correct) = 8.361594e-05
Pr(3 correct) = 2.787198e-05
Pr(4 correct) = 9.290660e-06
Pr(5 correct) = 3.096887e-06
Pr(6 correct) = 1.032296e-06
Pr(7 correct) = 3.440985e-07
Pr(8 correct) = 1.146995e-07
Pr(9 correct) = 3.823317e-08
Pr(10 correct) = 1.274439e-08
Pr(11 correct) = 4.248130e-09
Pr(12 correct) = 1.416043e-09
Pr(13 correct) = 4.720144e-10
Pr(14 correct) = 1.573381e-10
Pr(15 correct) = 5.244605e-11
Pr(16 correct) = 1.748202e-11
Pr(17 correct) = 5.827339e-12
Pr(18 correct) = 1.942446e-12
Pr(19 correct) = 6.474821e-13
Pr(20 correct) = 2.158274e-13
Pr(21 correct) = 7.194245e-14
Pr(22 correct) = 2.398082e-14
Pr(23 correct) = 7.993606e-15
Pr(24 correct) = 2.664535e-15
Pr(25 correct) = 8.881784e-16<br>


(a)  Adding the last 5 lines gives:
     P(more than 20 are correct) = {{{ 1.075x10^(-13) }}}<br>

(b)  Adding the first 5 lines gives:
     P(less than 5 are correct) = {{{ 1.124x10^(-3) }}}


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Hmmm,  I think I've made a mistake -- the sum of the table should be 1.000 (as one of the cases must certainly occur), but it is not.   I will investigate and post the fix later...
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================= Ignore solution above, CORRECTED below ==============<br>

Doh!  P(exactly r correct) = {{{ (1/4)^r*(3/4)^(25-r)*nCr }}} where n=25, r=number correct.  I will fix the table shortly...

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Corrected solution:

perl -e '$sum = 0; for($c=0; $c<26; $c++) { $ncr=1; for($i=25;$i>(25-$c);$i--) { $ncr *= $i; } for ($i=1; $i<=$c; $i++) { $ncr /= ($i);}  $p = (1/4)**($c)*(3/4)**(25-$c)*$ncr; printf "Pr(%d correct) = %e\n", $c, $p; $sum+=$p; } print "sum = $sum\n"; ' 
Pr(0 correct) = 7.525435e-04
Pr(1 correct) = 6.271195e-03
Pr(2 correct) = 2.508478e-02
Pr(3 correct) = 6.410555e-02
Pr(4 correct) = 1.175268e-01
Pr(5 correct) = 1.645376e-01
Pr(6 correct) = 1.828195e-01
Pr(7 correct) = 1.654082e-01
Pr(8 correct) = 1.240561e-01
Pr(9 correct) = 7.810941e-02
Pr(10 correct) = 4.165835e-02
Pr(11 correct) = 1.893561e-02
Pr(12 correct) = 7.363850e-03
Pr(13 correct) = 2.454617e-03
Pr(14 correct) = 7.013190e-04
Pr(15 correct) = 1.714335e-04
Pr(16 correct) = 3.571532e-05
Pr(17 correct) = 6.302704e-06
Pr(18 correct) = 9.337339e-07
Pr(19 correct) = 1.146691e-07
Pr(20 correct) = 1.146691e-08
Pr(21 correct) = 9.100720e-10
Pr(22 correct) = 5.515588e-11
Pr(23 correct) = 2.398082e-12
Pr(24 correct) = 6.661338e-14
Pr(25 correct) = 8.881784e-16<br>



(a)  Adding the last 5 lines gives:
     P(more than 20 are correct) = {{{ 9.677x10^(-10) }}}<br>

(b)  Adding the first 5 lines gives:
     P(less than 5 are correct) = {{{ 2.137x10^(-1) }}}