Represent the following situation using Venn diagrams and answer the questions that follow. In a survey of 100 students in the i
1 answer:
There are 80 students studying at least one language
<h3>The Venn Diagram</h3>The given parameters are:
- Italian = 28
- German = 30
- French = 42
- Italian and French = 10
- Italian and German = 8
- German and French = 5
- Three languages = 3
Using the above parameters, we have:
German and French only = 5 - 3 = 2
Italian and French only = 10 - 3 = 7
Italian and German only = 8 - 3 = 5
French only = 42 - 7 - 2 - 3 = 30
German only = 30 - 2 - 5 - 3 = 20
Italian only = 28 - 7 - 5 - 3 = 13
See attachment for the Venn diagram
<h3>Students studying none of the languages</h3>This is calculated as:
None + At least one = Total
Using the Venn diagram, we have:
None + (3 + 2 + 5 + 7 + 13 + 20 + 30) = 100
This gives
None + 80 = 100
Subtract 80 from both sides
None = 20
Hence, there are 20 students studying none of the languages
<h3>Students studying French only</h3>This is calculated as
French only = French - French and Italian - French and German + All three
So, we have:
French only = 42 - 10 - 5 + 3
Evaluate
French only = 30
Hence, there are 30 students studying French only
<h3>Students studying at least one</h3>In (b), we have:
None + At least one = Total ⇒ None + (3 + 2 + 5 + 7 + 13 + 20 + 30) = 100
By comparing both equations, we have
At least one = 3 + 2 + 5 + 7 + 13 + 20 + 30
This gives
At least one = 80
Hence, there are 80 students studying at least one language
Read more about Venn diagram at:
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Answer:
a) 0.3064 = 30.64% probability that athlete will test positive for prohibited drug use.
b) 0.2693 = 26.93% probability that this drug user never used the prohibited drug.
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
In which
P(B|A) is the probability of event B happening, given that A happened.
is the probability of both A and B happening.
P(A) is the probability of A happening.
A. If an athlete is selected at random, what is the probability that athlete will test positive for prohibited drug use?
100 - 5 = 95% of 8%
100 - 13 = 87% of 17%
11% of 75%
So
0.3064 = 30.64% probability that athlete will test positive for prohibited drug use.
B. If an athlete tests positive after a drug test, what is the probability that this drug user never used the prohibited drug?
Conditional Probability
Event A: Tests Positive
Event B: Never used the drug.
0.3064 = 30.64% probability that athlete will test positive for prohibited drug use
This means that
Probability of testing positive while never using the drug.
11% of 75%. So
0.2693 = 26.93% probability that this drug user never used the prohibited drug.