The probability that Chloe passes a test is 3/5.
The probability that Chloe and Danny pass the test is 7/15.
Calculate the probability that both Chloe and Danny fail the test.
It seems clear to me that you left out a key part of this problem.
Surely we were given that their passing the test are independent events.
If so, then so is failing the test independent.
If they are independent, the probability that Danny passes a test is 7/9.
So probability that Chloe fails a test is 2/5 and probability that Danny fails a test is 2/9. So what is the probability that both Chloe and Danny fail the test?
Hello, GAdams!
Assuming the events are independent, we have:Four students sit a typing test.
(a) The probability that Anna passes the test is $\displaystyle \frac{1}{2}$.
The probability that Boris passes the test is $\displaystyle \frac{4}{9}$.
Calculate the probability that both Anna and Boris pass the test.
. . $\displaystyle P(A \cap B) \;=\;P(A)\cdot P(B) \;=\;\left(\frac{1}{2}\right)\left(\frac{4}{9}\rig ht)\;=\;\boxed{\frac{2}{9}}$
We are given: .$\displaystyle P(C) = \frac{3}{5},\;P(C \cap D) = \frac{7}{15}$(b) The probability that Chloe passes the test is $\displaystyle \frac{3}{5}$.
The probability that Chloe and Danny pass the test is $\displaystyle \frac{7}{15}$.
Calculate the probability that both Chloe and Danny fail the test.
Since $\displaystyle P(C \cap D) \:=\:P(C)\cdot P(D)$
. . we have: .$\displaystyle \frac{7}{15} \:=\:\frac{3}{5}\!\cdot\!P(D)\quad\Rightarrow\quad P(D) \:=\:\frac{\frac{7}{15}}{\frac{3}{5}}\:=\:\frac{7} {9}$
Now we have: .$\displaystyle \begin{Bmatrix}P(C) = \frac{3}{5} & \Rightarrow & P(\overline{C}) = \frac{2}{5} \\ P(D) = \frac{7}{9} & \Rightarrow & P(\overline{D}) = \frac{2}{9}\end{Bmatrix}$
Therefore: .$\displaystyle P(\overline{C} \cap \overline{D}) \;=\;P(\overline{C})\cdot P(\overline{D}) \;=\;\left(\frac{2}{5}\right)\left(\frac{2}{9}\rig ht) \;=\;\boxed{\frac{4}{45}}$