A study of the US clinical population found that 24.1% are diagnosed with a ment
ID: 3240368 • Letter: A
Question
A study of the US clinical population found that 24.1% are diagnosed with a mental disorder, 14.9% are diagnosed with an alcohol-related disorder, and 5% are diagnosed with both disorders.
(a) What is the probability that someone from the clinical population is diagnosed with a mental disorder, knowing that the person is diagnosed with an alcohol-related disorder? Please use 3 decimal places.
(b) What is the probability that someone from the clinical population is diagnosed with an alcohol-related disorder, knowing that the person is diagnosed with a mental disorder? Please use 3 decimal places.
Explanation / Answer
Now, formulas for these sorts of things exist. However, I don't like to think of this through formulas but rather think through it using logic.
Secondly, I don't like working in percentages so I'm going to change all these figures into decimals by dividing them by 100 and changing them back into percentages at the end.
So, here's my explanation.
Basically, the chance of BOTH occuring at once is 5% or 0.05.
Now, in (a), given that they've already got a MENTAL disorder, they want us to find the chance he's going to have an ALCOHOL disorder. For 2 events to occur simultaneously, we apply the AND rule which means we multiply the 2 probabilites. So, he's already got a MENTAL disorder so that's 12.4% or 0.124 already there. That multiplied by SOMETHING (ie the chance of him having an alcohol disorder) should equal 0.05. So:
0.124 x a = 0.05
Rearrange:
a = 0.05/0.124 = 0.40323...
Multiply by 100 to make it a percentage
0.40323 * 100 = 40.323% to 3dp
Now, in (b), given that they've already got an ALCOHOL disorder. They want us to find the chance he's going to have a MENTAL disorder. Again, for 2 events to occur simultaneously, we apply the AND rule which means we multiply the 2 probabilites. So, he's already got an ALCOHOL disorder so that's 23.8% or 0.238 already there. That multiplied by SOMETHING (ie the chance of him having a mental disorder) should equal 0.05. So:
0.238 x a = 0.05
Rearrange:
a = 0.05/0.238 = 0.21008...
Multiply by 100 to make it a percentage
0.21008 * 100 = 21.008% to 3dp
I'm 95% sure this is correct.
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