Is he/she really innocent?

Probability Level 3

At a High Court, people on trial are judged as follows:

For people who really are guilty, the judgment says " Yes " 97% of the time.
For people who are in fact innocent, the judgment says " Yes " 4% of the time (" false positive ")

If 72.5% of people on trial really are guilty, and the judgment for a randomly selected person says " Innocent ", what are the probability that the person really is innocent?

92.39% 28.575% 98.9% 100% 26.4%

2 solutions

Munem Shahriar
Jul 23, 2017

	Judgement says '' Yes ''
Is guilty	$97\%$
Is innocent	$4\%$ '' false positive ''
	Judgement says '' No ''
Is guilty	$3\%$ '' false negative ''
Is Innocent	$96\%$

Now, drawing a diagram

First of all, let's check that all the percentages add up:

$\Rightarrow 70.325\% + 2.175\% + 1.1\% + 26.4\% = 100\%$ (good!)

And the two " No " answers add up to $\Rightarrow 2.175\% + 26.4\% = 28.575\%$ , but $26.4\%$ are correct.

$\dfrac{26.4}{28.575}$ $=$ $\color{#69047E} \boxed{92.39\%}$

Saya Suka
Apr 13, 2021

P(innocent | "not guilty" judgement)
= (27.5%)(100% – 4%) / [(72.5%)(100% – 97%) + (27.5%)(100% – 4%)]
= (0.275)(0.96) / [(0.725)(0.03) + (0.275)(0.96)]
= (11)(32) / [(29)(1) + (11)(32)]
= 352 / [29 + 352]
= 352 / 381
= 0.9238845
= 92.39%

Is he/she really innocent?

2 solutions

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