Practice Problems & Exam info

P(Fails in 5 | exceeded 10)

= P(Fails in 5 n exceeds 10)/P(exceeds 10)

= P(fails in 15 n exceeds 10) / 1-P(doesnt exceed 10)

Side: P(failing in 15) = P(fails in 15 n exceeds 10) + P(fails in 15 n not exceed 10)

P(fails in 15 n exceeds 10) = P(failing in 15) - P(failing in 15 n not exceed 10)

Answer: P(fails in 15) - P(fails in 10)/1 - P(fails in 10)

Suppose that in order for the defendant to be convicted in a jury trial, at least eight of the 12 jurors must enter a guilty vote. Assume each juror makes the correct decision with probability 0.7 independently of other jurors. If 40% of the defendants in such jury trials are innocent, what is the proportion of correct verdicts?

n = 12

need >= 8 to vote guilty to convict

need >= 5 to vote innocent for free

P(indiviudal votes correctly) = .7

P(someone is innocent) = .4

?P(a correct verdict)?

A = innocent

B = jurry reaches correct verdict

?P(B)?

P(B) = P(AnB) + P(A^cnB)

= P(A) * P(B|A) + P(A^c) * P(B|A^c)

= .4 * P(B|A) + .6 * P(B|A^c)

P(1 juror votes out of 12)

12 * .7 * .3¹¹

CWWWWwWW..W

?P(B|A)?

P(jury reaches correct verdict given they are innocent)

let X = # of correct votes

They have to be innocent so take the probability of all the cases that the jury reaches innocence.

x = 5

n = 12

r = 5 to 12

(12 choose r) * .7^r*(.3^(12-r))

?P(B|A^c)?

Probabilty that the jurry reaches correct verdict given they are not innocent